Cambridge A Level Mathematics - Further 9231 — 2025 May/June Paper 4 · Variant 1

This document has 16 pages. Any blank pages are indicated. [Turn over Cambridge International AS & A Level FURTHER MATHEMATICS 9231/41 Paper 4 Further Probability & Statistics May/June 2025 1 hour 30 minutes You must answer on the question paper. You will need: List of formulae (MF19) INSTRUCTIONS ● Answer all questions. ● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ● Write your name, centre number and candidate number in the boxes at the top of the page. ● Write your answer to each question in the space provided. ● Do not use an erasable pen or correction fluid. ● Do not write on any bar codes. ● If additional space is needed, you should use the lined page at the end of this booklet; the question number or numbers must be clearly shown. ● You should use a calculator where appropriate. ● You must show all necessary working clearly; no marks will be given for unsupported answers from a calculator. ● Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place for angles in degrees, unless a different level of accuracy is specified in the question. INFORMATION ● The total mark for this paper is 50. ● The number of marks for each question or part question is shown in brackets [ ]. * 4 4 5 3 7 3 9 1 2 1 * DC (ST) 355987 © UCLES 2025 ,










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3 9231/41/M/J/25 © UCLES 2025 [Turn over 1 The manager of a hardware store is interested in whether there is a difference in the amount spent per customer on weekdays ($x) compared to weekends ($y). Random samples of 120 customers on weekdays and 80 customers on weekends are taken and the amount spent by each customer is recorded. The results are summarised as follows. x 10 470 = / ( ) x x 12 283 2 - = / y 6560 = / ( ) y y 13520 2 - = / Test at the 1% significance level whether there is a difference in the mean amount spent per customer on weekdays compared to weekends. You should not assume that the population variances of the amounts spent on weekdays and weekends are equal. [7] … … … … … … … … … … … … … … … … … … … … … … … * 0000800000003 * ,










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4 9231/41/M/J/25 © UCLES 2025 2 x f(x) 5 8 0 As shown in the diagram, the continuous random variable X has probability density function f given by ( ) f x = , , otherwise, a x x x b c 0 0 5 5 8 G G G G - * where a, b and c are constants. (a) Show that a 13 2 = and find the values of b and c. [3] … … … … … … … … … … … … … … … … … * 0000800000004 * ,










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5 9231/41/M/J/25 © UCLES 2025 [Turn over (b) Find the mean of X. [3] … … … … … … (c) Find the median of X. [2] … … … … … The random variable Y is defined by Y X 2 = . (d) Find the cumulative distribution function for Y. [4] … … … … … … … … … … … … … … * 0000800000005 * ,










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6 9231/41/M/J/25 © UCLES 2025 3 Eggs in a supermarket are sold in boxes of six. A supermarket manager wishes to model the number of broken eggs in the boxes sold in the store. A random sample of 2000 boxes is taken and the number of broken eggs recorded. The observed frequencies are shown in the table below. Number of broken eggs 0 1 2 3 4 5 6 Observed frequency 1844 143 11 0 1 0 1 (a) Use the data to estimate the probability that an egg is broken. Give your answer correct to 4 significant figures. [1] … … … … … … It is decided to carry out a goodness of fit test at the 0.5% significance level to determine whether a binomial distribution fits the data. The observed frequencies and the expected frequencies are given in the following table. Number of broken eggs 0 1 2 3 4 5 6 Observed frequency 1844 143 11 0 1 0 1 Expected frequency 1831.3 a 6.016 0.119 0.001 0.000 0.000 (b) Show that . a 162 6 = correct to 1 decimal place. [1] … … … … … … … … … … * 0000800000006 * ,










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7 9231/41/M/J/25 © UCLES 2025 [Turn over (c) Carry out a goodness of fit test at the 0.5% level of significance to test whether a binomial distribution is a satisfactory model for the data. [5] … … … … … … … … … … … … … … … … … … … … … … … (d) Give a reason why a binomial distribution may not be a suitable model in this situation. [1] … … … * 0000800000007 * ,










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8 9231/41/M/J/25 © UCLES 2025 4 A researcher is interested in whether there is a difference between two schools in students’ aptitude for English. She randomly chooses 10 students from school X and 8 students of a similar age from school Y to take a written English test. The scores for the students from school X (x) and school Y (y) are summarised as follows. x 612 = / x 40104 2 = / y 444 = / y 27 460 2 = / You should assume that the two distributions are normal and have the same population variance. (a) Find a 95% confidence interval for the difference in the mean scores for students from school X and students from school Y in the written English test. [6] … … … … … … … … … … … … … … … … … … … … … … * 0000800000008 * ,










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9 9231/41/M/J/25 © UCLES 2025 [Turn over … … … … … … … … … … … … … … … … … … … … … … … (b) Use the confidence interval you found in part (a) to explain why there is insufficient evidence at a 5% significance level to suggest that the English scores of students from school X and students from school Y are different. [1] … … … * 0000800000009 * ,










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10 9231/41/M/J/25 © UCLES 2025 5 Two jigsaw puzzles have the same number of pieces with identical shapes but have different pictures printed on them. One puzzle has a seaside picture and the other has a cartoon picture. A researcher believes that children will complete the cartoon puzzle more quickly. To test this belief, 10 children are randomly selected. The time taken in seconds for each child to complete each puzzle is recorded below. Child A B C D E F G H I J Seaside 182 130 193 181 192 204 184 192 180 189 Cartoon 161 111 195 159 202 200 168 165 145 160 (a) Carry out a Wilcoxon matched-pairs signed-rank test at the 5% significance level to test the researcher’s belief. [6] … … … … … … … … … … … … … … … … … … … … … * 0000800000010 * ,










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11 9231/41/M/J/25 © UCLES 2025 [Turn over (b) Show that using a paired-sample sign test at the 5% significance level would result in the opposite conclusion to that found in part (a). [3] … … … … … … … … … … … … … … … … … … … … … It was later discovered that the experiment had been conducted such that each child completed the seaside puzzle first followed by the cartoon puzzle. (c) Comment on the validity of using this experiment to test the researcher’s belief. [1] … … … * 0000800000011 * ,









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12 9231/41/M/J/25 © UCLES 2025 6 Y is a discrete random variable which takes the values 0, 2, 4, … . The probability generating function of Y is given by ( ) G t at k 1 Y 2 = - . (a) Find k in terms of a. [1] … … … … … … (b) Show that ( ) Y a 2 P 2 2 = . [3] … … … … … … … … … … … … … … … … … … * 0000800000012 * ,









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13 9231/41/M/J/25 © UCLES 2025 It is now given that . a 0 2 = . (c) Find the value of E(Y ). [2] … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000013 * ,









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14 9231/41/M/J/25 © UCLES 2025 Additional page If you use the following page to complete the answer to any question, the question number must be clearly shown. … … … … … … … … … … … … … … … … … … … … … … … … … … * 0000800000014 * ,









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16 9231/41/M/J/25 © UCLES 2025 Permission to reproduce items where third-party owned material protected by copyright is included has been sought and cleared where possible. Every reasonable effort has been made by the publisher (UCLES) to trace copyright holders, but if any items requiring clearance have unwittingly been included, the publisher will be pleased to make amends at the earliest possible opportunity. To avoid the issue of disclosure of answer-related information to candidates, all copyright acknowledgements are reproduced online in the Cambridge Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge. BLANK PAGE * 0000800000016 * ,









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This document consists of 16 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge International AS & A Level FURTHER MATHEMATICS 9231/41 Paper 4 Further Probability & Statistics May/June 2025 MARK SCHEME Maximum Mark: 50 Published This mark scheme is published as an aid to teachers and candidates, to indicate the requirements of the examination. It shows the basis on which Examiners were instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began, which would have considered the acceptability of alternative answers. Mark schemes should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the May/June 2025 series for most Cambridge IGCSE, Cambridge International A and AS Level components, and some Cambridge O Level components.

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 2 of 16 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptions for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently, e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 3 of 16 Mathematics Specific Marking Principles 1 Unless a particular method has been specified in the question, full marks may be awarded for any correct method. However, if a calculation is required then no marks will be awarded for a scale drawing. 2 Unless specified in the question, non-integer answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the degree of accuracy is not affected. 3 Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points. 4 Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (ISW). 5 Where a candidate has misread a number or sign in the question and used that value consistently throughout, provided that number or sign does not alter the difficulty or the method required, award all marks earned and deduct just 1 A or B mark for the misread. 6 Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear.

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 4 of 16 Annotations guidance for centres Examiners use a system of annotations as a shorthand for communicating their marking decisions to one another. Examiners are trained during the standardisation process on how and when to use annotations. The purpose of annotations is to inform the standardisation and monitoring processes and guide the supervising examiners when they are checking the work of examiners within their team. The meaning of annotations and how they are used is specific to each component and is understood by all examiners who mark the component. We publish annotations in our mark schemes to help centres understand the annotations they may see on copies of scripts. Note that there may not be a direct correlation between the number of annotations on a script and the mark awarded. Similarly, the use of an annotation may not be an indication of the quality of the response. The annotations listed below were available to examiners marking this component in this series. Annotations Annotation Meaning More information required Accuracy mark awarded zero Accuracy mark awarded one Independent accuracy mark awarded zero Independent accuracy mark awarded one Independent accuracy mark awarded two Benefit of the doubt Blank Page Incorrect Dep Used to indicate DM0 or DM1

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 5 of 16 Annotation Meaning DM1 Dependent on the previous M1 mark(s) Follow through Indicate working that is right or wrong Highlighter Highlight a key point in the working Ignore subsequent work Judgement Judgement Method mark awarded zero Method mark awarded one Method mark awarded two Misread Omission or Other solution Off-page comment Allows comments to be entered at the bottom of the RM marking window and then displayed when the associated question item is navigated to. On-page comment Allows comments to be entered in speech bubbles on the candidate response. Judgment made by the PE Premature approximation Special case

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 6 of 16 Annotation Meaning Indicates that work/page has been seen Error in number of significant figures Correct Transcription error Correct answer from incorrect working

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 7 of 16 Mark Scheme Notes The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method mark, awarded for a valid method applied to the problem. Method marks are not 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. Correct application of a formula without the formula being quoted obviously earns the M mark and in some cases an M mark can be implied from a correct answer. 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). B Mark for a correct result or statement independent of method marks. DM or DB When a part of a question has two or more ‘method’ steps, the M marks are generally independent unless the scheme specifically says otherwise; and similarly, when there are several B marks allocated. The notation DM or DB is used to indicate that a particular M or B mark is dependent on an earlier M or B (asterisked) mark in the scheme. When two or more steps are run together by the candidate, the earlier marks are implied and full credit is given. FT Implies that the A or B mark indicated is allowed for work correctly following on from previously incorrect results. Otherwise, A or B marks are given for correct work only. • A or B marks are given for correct work only (not for results obtained from incorrect working) unless follow through is allowed (see abbreviation FT above). • For a numerical answer, allow the A or B mark if the answer is correct to 3 significant figures or would be correct to 3 significant figures if rounded (1 decimal place for angles in degrees). • The total number of marks available for each question is shown at the bottom of the Marks column. • Wrong or missing units in an answer should not result in loss of marks unless the guidance indicates otherwise. • Square brackets [ ] around text or numbers show extra information not needed for the mark to be awarded.

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 8 of 16 Abbreviations AEF/OE Any Equivalent Form (of answer is equally acceptable) / Or Equivalent AG Answer Given on the question paper (so extra checking is needed to ensure that the detailed working leading to the result is valid) CAO Correct Answer Only (emphasising that no ‘follow through’ from a previous error is allowed) CWO Correct Working Only ISW Ignore Subsequent Working SOI Seen Or Implied SC Special Case (detailing the mark to be given for a specific wrong solution, or a case where some standard marking practice is to be varied in the light of a particular circumstance) WWW Without Wrong Working AWRT Answer Which Rounds To

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 9 of 16 Question Answer Marks Guidance 1 0 1 H : H : x y x y     =  B1 If in words, must contain ‘population means’. Allow 0 1 H : 0, H : 0 d d   =  . 2 2 103.218... 171.139... x y s s = = M1 SOI, one unbiased estimate correct to 3sf or better. Do not accept biased estimates (102.4, 169.0).   2 2 2.999 120 80 y x s s + = A1 A0 for pooled s . Accept 3, 3.0, 3.00. May be implied by 3.03 = z .   10470 6560 120 80 2 2 3.03 120 80 y x z s s − = = + M1 Value of z obtained using correct form with their 2 2 and x y t eir s h s . Use of pooled s scores M0 A0. A1 Use of pooled s scores M0 A0. '3.03' > 2.576, reject 0 H (significant). OR ( ) 0 P '3.03' 0.00122 0.005, reject H Z  =  (significant). M1 Compare their z with 2.576 and appropriate conclusion (may be in terms of 1 H ). Reject H0 can be implied by an attempt at an appropriate conclusion in context. May see 0 0.00244 0.01, reject H  (significant). Allow M1 for comparison of their 3.03 with 2.326 and appropriate conclusion. There is sufficient evidence to suggest the mean spend of customers is different on weekends compared to weekdays. A1 Correct conclusion in context from correct working ignoring hypotheses. Level of uncertainty in language used (for example, not ‘prove’). Must refer to mean or average spend. 7

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 10 of 16 Question Answer Marks Guidance 2(a) 3 2 5 1 a a + = , leading to 2 13 a = . B1 AG, shown convincingly. Uses area = 1, b and c must be eliminated. Attempt to solve 2 independent equations in b and c or 3 independent equations in a , b and c . M1 May see any of the following: 39 3 2 2 39 2 3 , 5 , 8 0, , 5 3 1. 5 8 b c a a b c b c a c a b c − = = − − = −= + − = − OR equation of line through ( ) 5,a and ( ) 8,0 . 16 2 39 39 , b c = = A1 Correct values imply M1 A1. 3 2(b) ( ) ( ) 5 8 16 2 2 13 39 39 0 5 E d d X x x x x x  = + −    M1 Attempt to integrate f( ) x x with their b and c , limits seen. May be in terms of , , a b c . 5 8 2 2 3 8 1 2 13 39 117 0 5 x x x     + −     M1 Correct integrated expression with their b and their c , including limits. May be in terms of , , a b c . 43 13 3.31 = A1 AWRT 3.31. 3 2(c) 2 13 0.5 m = OR 2 13 0 d 0.5 m x=  M1 Uses 0 5 m   and area = 0.5, may be in terms of a . 13 4 3.25 m = = A1 2

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 11 of 16 Question Answer Marks Guidance 2(d) ( ) 2 16 2 1 13 39 39 , x x x k − + B1 FT Correct expressions for integrating both pieces of PDF, seen anywhere, using their b and c . 25 39 k = − B1 FT FT their b and their c to obtain k . ( ) 2 13 16 25 1 39 39 39 0 0 0 25 G 25 64 1 64 y y y y y y y y     =  − −      ≤ ≤ ≤ M1 Both parts with y substituted (condone missing k ), ignore limits. A1 FT FT their b , their c and their k . Fully defined including correct domain in terms of y . Domain must cover all reals, may overlap at boundaries. 4 Question Answer Marks Guidance 3(a) 0 1844 1 143 2 11 3 0 4 1 5 0 6 1 7 2000 6 480  +  +  +  + +  +    =      B1 175 12000 , AWRT 0.01458. 1 3(b) 2000 (1831.3 6.016 0.119 0.001) 162.6 a = − + + + = B1 AG, shown convincingly. Alternative method for question 3(b) ( ) 5 7 7 480 480 6 1 2000 1 a  =   −    = 162.6 B1 AG, shown convincingly. 1

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 12 of 16 Question Answer Marks Guidance 3(c) H0 : A binomial distribution is a satisfactory model for the data. H1 : A binomial distribution is not a satisfactory model for the data. B1 Must mention binomial and data. ( ) ( ) ( ) 2 2 2 1844 1831.3 143 162.6 13 6.136 1831.3 162.6 6.136 − − − + +   0.08807 2.3626 7.6784 = + + OR 2 2 2 1844 143 13 2000 1831.3 162.6 6.136 + + − M1 Expression of correct form with cells correctly combined. May use 6.1 (found by subtraction). Implied by 2 correct contributions. 10.1 A1 AWRT 10.1. Allow AWRT 10.3 from using 6.1. SC unsupported 10.1 (or 10.3) scores M0 B1. [1 degree of freedom so critical value is 7.879.] '10.1' 7.879,  reject 0 H (significant). M1 Compare their 10.1 (or 10.3) with 7.879 and appropriate conclusion (may be in terms of H1). Reject H0 can be implied by an attempt at an appropriate conclusion in context. Allow M1 for comparison of their 10.1 (or 10.3) with 10.6 and appropriate conclusion. Sufficient evidence to suggest that the manager’s model is not satisfactory. A1 Correct conclusion in context from correct working ignoring hypotheses. Level of uncertainty in language used (for example, not ‘prove’). 5 3(d) Not independent or probability of “success” may not be constant or plausible reason in context (for example, higher number of boxes with more than 1 broken egg due to boxes being dropped). B1 Credit any valid comment in context. 1

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 13 of 16 Question Answer Marks Guidance 4(a) ( )   2 2 1 10 40104 612 2649.6 x x  − = −  = ( )   2 2 1 8 27460 444 2818 y y  − = − = M1 Either expression, may be embedded. May be implied, for example, by 294.4 or 402.57 seen.   2 2649.6 2818 341.725 10 8 2 + = = + − ps A1 SOI ( )  2 1 1 10 8 (61.2 55.5) 2.120 ' ' 5.7 2.12 8.7686 ps −  + =   B1 2.120, SOI. M1 Correct form, must be a t value. A1 Correct un-simplified confidence interval, SOI.   12.9, 24.3 − A1 Accept with inequality signs or open brackets or   24.3, 12.9 − from − Y X . Condone   24.3, 12.9 − . Do not accept 5.7 18.6  . 6 4(b) Because zero lies inside the confidence interval. B1 1

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 14 of 16 Question Answer Marks Guidance 5(a) 0 H : = s c m m 1 H :  s c m m B1 If in words, must contain ‘population median’. Allow H0 : md = 0, H1 : md > 0 (or < 0) if defined or consistent with working. d 21 19 –2 22 –10 4 16 27 35 29 Rank 6 5 –1 7 –3 2 4 8 10 9 M1 Attempt at differences. M1 Attempt at (signed) ranks. ( 51, 4) 4 P Q T = = = A1 Allow errors in positive ranks. '4' 10, ≤ reject 0 H (significant). M1 Compare their 4 (from rank sum only) with 10 & appropriate conclusion (allow in terms of H1). Reject H0 can be implied by an attempt at an appropriate conclusion in context. Allow M1 for comparison of their 4 (from rank sum only) with 8 and appropriate conclusion. Sufficient evidence to suggest that children, on average, complete the cartoon puzzle more quickly than the seaside puzzle. OR Sufficient evidence to support the researcher’s belief. A1 CWO ignoring hypotheses. In context, level of uncertainty in language. A0 for 4 T = from incorrect ranks/differences. 6 5(b) ( ) ~ B 10, 0.5     X ( ) P 2 0.0547 X = ≤ B1 OR ( ) P 8 0.0547 X = ≥ . '0.0547' 0.05  , accept H0 (not significant). M1 Compare their 0.0547 with 0.05 and appropriate conclusion (may be in terms of H1). In part 5(a) H0 is rejected, in part 5(b) H0 is accepted, so opposite conclusion reached. A1 CWO, compares with test result in part 5(a). 3 5(c) The order of the tests means that children may do the second test quicker (or slower) because, for example, they have had practice (or are tired). B1 Credit any valid comment in context. 1

9231/41 Cambridge International AS & A Level – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 15 of 16 Question Answer Marks Guidance 6(a) 1 = − k a B1 1 6(b) ( ) ( ) 1 2 2 1 1 − − = + + at at *M1 Attempt at series for ( ) 1 2 1 − −at with 2 1+ at seen or implied. ( ) ( ) P 2 1 1  = − + Y k a DM1 Correct un-simplified expression in terms of their k and/or a . ( ) ( )( ) 2 P 2 1 1 1  = − − + = Y a a a A1 AG, shown convincingly. Alternative method for question 6(b) ( ) ( ) ( ) ( ) ( ) 1 1 2 2 ' '' P 0 G 0 ( 1) G 0 0 P 2 G 0 (2 ). Y Y Y Y k P Y Y ak   = = = = = =   = = = *M1 Attempt at ( ) '' G 0 Y with their k . ( ) P 2 1 Y k ak  = − − DM1 Correct un-simplified expression in terms of their k and/or a . ( ) ( ) ( ) 2 P 2 1 1 1 Y a a a a  = − − − − = A1 AG, shown convincingly. 3 6(c) ( ) ( ) 2 2 ' 0.32 G 1 0.2 Y t t t = − M1 Attempt to differentiate ( ) GY t with their k , may be in terms of a and/or k . ( ) ( ) ( ) 2 ' 0.32 E G 1 0.5 1 0.2 Y Y = = = − A1 CWO 2

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