Cambridge IGCSE Mathematics - International 0607 — 2025 May/June Paper 6 · Variant 1

This document has 12 pages. [Turn over * 3 3 9 1 1 5 9 3 9 3 * Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 Investigation and Modelling (Extended) May/June 2025 1 hour 30 minutes You must answer on the question paper. No additional materials are needed. 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. ● You should use a graphic display calculator where appropriate. ● You may use tracing paper. ● You must show all necessary working clearly, including sketches, to gain full marks for correct methods. ● In this paper you will be awarded marks for providing full reasons, examples and steps in your working to communicate your mathematics clearly and precisely. INFORMATION ● The total mark for this paper is 50. ● The number of marks for each question or part question is shown in brackets [ ]. DC (DE/SG) 346945/3 © UCLES 2025 ,










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2 0607/61/M/J/25 © UCLES 2025 Section A INVESTIGATION PRODUCTS OF PAIRS You are advised to spend no more than 45 minutes on this section. In this investigation you will look at the differences between the products of pairs of numbers in an increasing linear sequence of positive numbers. In each sequence: • there are 3 or more terms • the difference between each term is the increase • the first term and the last term are the outside pair • the terms next to the first and last terms are the inside pair. To find the product difference multiply the numbers in each pair and find the difference. Example The sequence is inside pair 2 3 outside pair 1 4 Product of inside pair 2 3 6 # = Product of outside pair 1 4 4 # = Product difference 6 4 2 - = 1 The sequences in this question have 4 terms. (a) Complete the table. Sequence Increase Product of Product difference inside pair outside pair 1, 2, 3, 4 1 6 4 2 6, 8, … , 12 … … 72 … 7.2, … , … , 16.2 … … 116.64 18 … , … , 96, 100 … … … … … , … , … , … 5 6000 … … [5] * 0000800000002 * ,










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3 0607/61/M/J/25 © UCLES 2025 [Turn over (b) (i) Complete the table. Use your answers in part (a) to help you. Sequences with 4 terms Increase, n Product difference 1 2 2 3 18 4 5 6 [2] (ii) Find an expression, in terms of n, for the product difference. … [2] (iii) A sequence of 4 terms has an increase of 11. Use your expression in part (b)(ii) to find the product difference for this sequence. … [2] (iv) Use a sequence of numbers to show that your answer to part (b)(iii) is correct. [1] * 0000800000003 * ,










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4 0607/61/M/J/25 © UCLES 2025 2 (a) A sequence has 3 terms. The multiplication of the inside pair is the middle number squared. Example The sequence is 1 2 3 Product of inside pair 2 2 2 4 2 # = = Product of outside pair 1 3 # = 3 Product difference 4 3 - = 1 (i) Complete the table. Use your own sequences to help you. Sequences with 3 terms Increase, n Product difference 1 1 2 3 4 [2] (ii) Write down an expression, in terms of n, for the product difference. … [1] * 0000800000004 * ,










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5 0607/61/M/J/25 © UCLES 2025 [Turn over (b) A sequence has 5 terms. The inside pair is the two terms that are one away from each end. Example The sequence is inside pair 2 4 3 1 5 Product of inside pair 2 4 8 # = Product of outside pair 1 5 5 # = Product difference 8 5 3 - = (i) Complete the table. Use your own sequences to help you. Sequences with 5 terms Increase, n Product difference 1 3 2 3 4 [3] (ii) A sequence with 5 terms has a product difference of 432. An expression for the product difference is n 3 2. Find the increase for this sequence. … [1] * 0000800000005 * ,










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6 0607/61/M/J/25 © UCLES 2025 (c) (i) Complete the table. Use your answers to Question 1(b)(ii) and Question 2(a)(ii) to help you. Number of terms in sequence, x Product difference 3 4 5 n 3 2 6 [1] (ii) Find an expression, in terms of x and n, for the product difference. … [1] (d) The number of terms in a sequence is equal to the increase. Find a sequence with a product difference of 1859. Write down the first and last terms of your sequence. First term = … Last term = … [4] * 0000800000006 * ,










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7 0607/61/M/J/25 © UCLES 2025 [Turn over Section B MODELLING CROSSED POLES You are advised to spend no more than 45 minutes on this section. In this task you will be modelling the heights of triangles made below two poles when they cross. Each triangle is made from 2 poles that lean between vertical posts. h d ground post pole One end of each pole rests on the ground and the other end rests against a vertical post. The posts are d metres apart on horizontal ground. The poles cross each other h metres above the ground. In this task you will find Pythagoras’ theorem useful. a c b a b c 2 2 2 + = * 0000800000007 * ,










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8 0607/61/M/J/25 © UCLES 2025 3 The poles are 10 m long. Each pole touches one post on the ground and the other post a metres above the ground. d NOT TO SCALE a P X Y Z Q h Q is the midpoint of ZY. (a) (i) Complete the statement. P Triangle … is similar to triangle XYZ. [1] (ii) Use similar triangles and a and h to complete this statement. d d 2 1 = [1] (b) In this part h 4 = . (i) Write down the value of a. … [1] (ii) Use Pythagoras’ theorem to find the value of d. … [2] * 0000800000008 * ,










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9 0607/61/M/J/25 © UCLES 2025 [Turn over (c) (i) Show that a model for h in terms of d is h d 2 1 100 2 = - . [2] (ii) Sketch the model for h. h d 0 0 Distance between the posts (metres) Height of the poles crossing (metres) [2] (d) Look at the diagram on page 8. (i) Describe the position of the poles when the curve meets the d-axis on the graph. … [1] (ii) Give a reason why the model is not valid when the curve meets the h-axis on the graph. … [1] * 0000800000009 * ,










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10 0607/61/M/J/25 © UCLES 2025 4 Two poles are each 15 m long. (a) Change the model in Question 3(c)(i) for poles of length 15 m. … [1] (b) The poles cross at a height of 6 m. Find the distance between the posts. … [3] 5 In this question the lengths of the two poles are different. One pole is 20 m long and the other pole is 15 m long. The 20 m pole touches the left post at a height of b metres above the ground. The 15 m pole touches the right post at a height of c metres above the ground. The poles cross at R, which is x metres from the right post. d x NOT TO SCALE b R K G F E S c h (a) (i) Use similar triangles to complete these statements. In triangles RSG and EFG d x = h . In triangles RSF and KGF d d x - = . [1] * 0000800000010 * ,










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11 0607/61/M/J/25 © UCLES 2025 [Turn over (ii) Use the equations in part (a)(i) to show that b h c h 1 + = . [1] (iii) Use the equation in part (a)(ii) to show that d h d h 400 1 225 2 2 - + - = . [3] (iv) Rearrange the model in part (a)(iii) to show that a model for h in terms of d is h d d d d 225 400 400 225 2 = - + - - - 2 2 2 ` ` `j j j . [1] Questions 5(a)(v), (b)(i) and (b)(ii) are printed on the next page. * 0000800000011 * ,









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12 0607/61/M/J/25 © UCLES 2025 (v) Sketch the model for h in part (a)(iv) for values of d between 0 and 15. h d 0 15 0 Distance between the posts (metres) Height of the poles crossing (metres) [1] (b) The distance between the posts is 8 m. (i) Find the height where the poles cross. … [1] (ii) Find the heights of the points where the poles touch the posts. … … [2] 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. * 0000800000012 * ,









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This document consists of 12 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) 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.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 2 of 12 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.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 3 of 12 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 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.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 4 of 12 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 Accuracy mark awarded two Accuracy mark awarded three Independent mark awarded zero Independent mark awarded one Independent mark awarded two Independent mark awarded three Benefit of the doubt Communication mark Incorrect Follow through Highlighter Highlight a key point in the working Ignore subsequent work Method mark awarded zero Method mark awarded one Method mark awarded two

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 5 of 12 Annotation Meaning Method mark awarded three Misread Omission 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. Premature rounding/approximation Special case Indicates that work/page has been seen Transcription error Correct Correct answer from incorrect working

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 6 of 12 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 marks, awarded for a valid method applied to the problem. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. B Mark for a correct result or statement independent of Method marks. 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 M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working nfww not from wrong working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 7 of 12 Question Answer Marks Partial Marks Investigation: Products of pairs 1(a) Sequence Increase Product of Product difference Inside pair Outside pair 1 2 3 4 1 6 4 2 6 8 10 12 2 80 72 8 7.2 10.2 13.2 16.2 3 134.64 116.64 18 88 92 96 100 4 8832 8800 32 70 75 80 85 5 6000 5950 50 4 B1 for each row One calculation to justify the required terms of a sequence or multiplication of a correct pair or subtraction of (their) outer and (their) inner pairs or in row three: 116.64 + 18 = 134.64 or for last row: 6000 or n2 + 5n – 6000 [= 0] or correct use of quadratic formula or sketches for y = x(x + 5) and y = 6000 or sketch of y = x2 + 5x – 6000 C1 1(b)(i) 72 in row 6 1 Three of the differences 6, 10, 14, 18, 22 seen or calculation of product difference for their 4-term sequence with increase 6 C1 1(b)(ii) 2n2 oe mark final answer 1 Three common second differences of 4 seen or square numbers 1, 4, 9, [16] seen C1 1(b)(iii) 2  112 C1 FT their algebraic expression 242 1 1(b)(iv) Any 4-term sequence with increase of 11 and 2nd term  3rd term – 1st term  4th term = 242 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 8 of 12 Question Answer Marks Partial Marks 2(a)(i) Increase, n Product difference 1 1 2 4 3 9 4 16 1 One 3-term sequence seen and products and difference shown C1 2(a)(ii) n2 oe mark final answer 1 2(b)(i) Increase, n Product difference 1 3 2 12 3 27 4 48 1 Two 5-term sequences seen from a sequence with increase of 2, a sequence with increase of 3, a sequence with increase of 4 and for one sequence, products and difference shown C2 C1 for one 5-term sequence seen and products and difference shown 2(b)(ii) 12 1 2(c)(i) 4n2 oe in row 4 1 2(c)(ii) (x – 2)n2 oe 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 9 of 12 Question Answer Marks Partial Marks 2(d) Sketches of y = (x – 2)x2 and y = 1859 or sketch of y = x3 – 2x2 – 1859 or table of values or 3 1859 C1 [x =] or [n =] 13 1 1st term + (their13 – 1)  their13 or sequence written out or expression for nth term for their sequence C1 FT their 13 providing it is a stated integer such that 11⩽ n ⩽ 15 Correct first and last terms 1 FT their 13 providing it is a stated integer such that 11⩽ n ⩽ 15 Modelling: Crossed poles 3(a)(i) [P] QZ or [P] QY 1 3(a)(ii) h a 1 3(b)(i) 8 1 3(b)(ii) 2 2 2 8 10 d + = or better or 2 2 2 1 4 5 2 d  + =     or better C1 FT their 8 providing 4 < their 8 < 10 or their 5 providing 4 < their 5 <10 and their 5 is not 8 6 1 3(c)(i) 2 2 2 1 5 2 h d   + =     oe or ( ) 2 2 2 2 10 h d + = oe or 2 2 2 10 a d + = oe and 1 2 d h d a = oe stated or used at some point M1 Correct completion to given answer 2 1 100 2 h d = − A1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 10 of 12 Question Answer Marks Partial Marks 3(c)(ii) Correct sketch h d 0 0 1 Correct shape, meeting both axes Correct indication of scale for both axes C1 3(d)(i) Valid description: The poles are lying on the ground oe 1 3(d)(ii) Valid reason: There is no crossing point or There is no gap between the posts or The poles are vertical oe 1 4(a) 2 1 225 2 h d = − mark final answer 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 11 of 12 Question Answer Marks Partial Marks 4(b) Sketch of 2 1 225 2 h d = − and h = 6 and point of intersection indicated on diagram or graphs labelled or sketch of 2 1 225 2 h d = − – 6 and point of intersection with the d-axis indicated on diagram or graph labelled C2 C1 for sketch of 2 1 225 2 h d = − or 2 1 225 2 h d = − – 6 OR 2 1 6 225 2 d = − or 2 12 225 d = − and rearrangement to 2 2 225 12 d = − or 2 225 12 − or rearrangement of model 2 1 225 2 h d = − to ( ) 2 2 225 2 d h = − or   2 225 (2 ) d h = − and substitution of h = 6 OR C1 for correct substitution of h = 6 into 2 1 225 2 h d = − or 2 2 225 h d = − or their model in terms of h and d in 4(a) or rearrangement of model 2 1 2 h k d = − to ( ) 2 2 2 d k h = − or   2 (2 ) d k h = − where k is a positive constant OR diagram with 6 and 12 marked or 6 × 2 and 2 2 225 12 d = − or 2 2 15 12 − or diagram with 6 and 7.5 marked or 15  2 and 2 1 2 d       = 2 2 7.5 6 − or 2 2 7.5 6 − OR C1 for either diagram or 6 × 2 or 15  2 or d 2 = 225 – 122 or 2 2 15 12 − or 2 1 2 d       = 2 2 7.5 6 − or 2 2 7.5 6 − OR 2 2 7.5 6 − and 4.5 × 2 or 6 × 2 and 2 2 15 12 − OR C1 for 2 2 7.5 6 − or 4.5 × 2 or 6 × 2 or 2 2 15 12 − 9 [m] 1 5(a)(i) h b h c 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 12 of 12 Question Answer Marks Partial Marks 5(a)(ii) h h b c   + =     x d x d d − + and correct completion to 1 1 dep on (a)(i) correct 5(a)(iii) 2 2 2 20 b d + = oe 2 2 2 15 c d + = oe M2 M1 for each equation or for 2 2 20 b d = − and 2 2 15 c d = − 2 400 b d = − and 2 225 c d = − or 2 2 2 2 1 20 15 h h d d + = − − leading to 2 2 1 400 225 h h d d + = − − A1 5(a)(iv) 2 2 2 2 225 400 1 400 225 h d h d d d − + − = − − or 2 2 1 1 1 400 225 h d d   + =     − −   and 2 2 2 2 225 400 1 400 225 d d h d d   − + −  =   − −   OR 2 2 2 2 225 400 400 225 h d h d d d − + − = − − leading to given answer 2 2 2 2 400 225 225 400 d d h d d − − = − + − 1 5(a)(v) Correct sketch meeting d axis at 15 h d 0 0 15 1 5(b)(i) 7.5[0] m 1 5(b)(ii) 18.3 [m] 12.7 [m] 2 B1for each