Cambridge IGCSE Mathematics - International 0607 — 2025 May/June Paper 5 · Variant 1

This document has 8 pages. [Turn over * 7 6 4 8 8 8 6 1 3 9 * Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 Investigation (Core) May/June 2025 1 hour 15 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 40. ● The number of marks for each question or part question is shown in brackets [ ]. DC (DE/SG) 346944/3 © UCLES 2025 ,










, * 0000800000001 * ¬Oz> 4mHuOªEŠ`y5€W ¬ 4|O§•Œ\‡l¡˜cƒy‚ ¥E•UuuE5•U5 Eue•uU

2 0607/51/M/J/25 © UCLES 2025 INVESTIGATION PRODUCTS OF PAIRS 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 4 terms • the difference between each term is the increase • the first term and the last term are the outside pair • the middle two 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 (a) (i) The sequence is 5, 6, 7, 8. Complete the statements. Product of inside pair … 6 42 # = Product of outside pair … 5 40 # = Product difference 42 40 - = 2 [2] * 0000800000002 * ,










, ĬÍú¾Ġ´íÈõÏĪÅĊÝû·þ× Ĭą²ûÎģđĞãòñÇÙĞē»ñĂ ĥĕąÕµĕąÕåĕĥÅŵÅÕÕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN

3 0607/51/M/J/25 © UCLES 2025 [Turn over (ii) Each sequence in this question has 4 terms. In these sequences the increase is 1. Complete the table. Sequence Product of Product difference inside pair outside pair 1, 2, 3, 4 2 # 3 1 # 4 = 6 = 4 6 - 4 = 2 10 11, 12, 13 11 # 12 10 # 13 = … = 130 … - 130 = … 16, 17, … , … 17 # … 16 # … = 306 = 304 306 - 304 = 2 … , 25, 26, … 25 # 26 … # … = 650 = … 650 - … = … … ,… ,… , 101 … # … … # 101 = … = … … - … = … [6] (b) (i) Show that the product difference for this sequence is 2. 499, 500, 501, 502 [2] (ii) Complete this sequence of decimals with an increase of 1. Find the product difference. 2.5, 3.5, … , … Product difference … [3] * 0000800000003 * ,










, ĬÏú¾Ġ´íÈõÏĪÅĊÝù·þ× Ĭą±üÖĥčĎÖĈĀĂýĖÇ»āĂ ĥĕõĕõõĥµµĥµÅÅÕåĕÅÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN

4 0607/51/M/J/25 © UCLES 2025 (c) (i) Complete the statement. For a sequence of … terms the product difference is always … [1] (ii) These are expressions for the 4 terms of a sequence. a, a 1 + , a 2 + , a 3 + where a can be any number Show algebraically that your statement in part (c)(i) is true. [4] * 0000800000004 * ,










, ĬÍú¾Ġ´íÈõÏĪÅĊßû·Ā× Ĭą±ùÖğğěáĊćĉߺĥëùĂ ĥååĕµõĥĕÕÅÅÅąÕąĕÕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN

5 0607/51/M/J/25 © UCLES 2025 [Turn over 2 Each sequence in this question has 4 terms. In these sequences the increase is more than 1. (a) (i) Complete the table. Sequence Increase Product of Product difference inside pair outside pair 1, 3, 5, 7 2 15 7 8 6, 8, … , 12 2 … 72 … 7.2, 10.2, … , … … 134.64 … 18 20, … , 28, … 4 … 640 … 88, … , … , 100 … … 8800 … [5] (ii) A sequence has an increase of 5. The product of the inside pair is 6000. Find the 4 terms in the sequence. … , … , … , … [3] * 0000800000005 * ,










, ĬÏú¾Ġ´íÈõÏĪÅĊßù·Ā× Ĭą²úÎĩģīØðúÀû±ëĉĂ ĥåÕÕõĕąõŵĕÅąµĥÕÅÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN

6 0607/51/M/J/25 © UCLES 2025 (b) (i) Complete the table. Use your answers in part (a) to help you. Increase, n Product difference 1 2 2 8 3 18 4 5 50 6 [2] (ii) Find an expression, in terms of n, for the product difference. … [2] * 0000800000006 * ,










, ĬÑú¾Ġ´íÈõÏĪÅĊÞù¶þ× Ĭą±úÑģċġÐûċµěºÔăùĂ ĥõõÕõµąÕąÕµÅŵÅĕĕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN

7 0607/51/M/J/25 © UCLES 2025 [Turn over (c) (i) A sequence has an increase of 11. Use your expression in part (b)(ii) to find the product difference for this sequence. … [2] (ii) The first term in a sequence is 12. The sequence has an increase of 11. Write down the sequence. 12, … , … , … [1] (iii) The first term in a sequence is a. The sequence has an increase of 11. Complete the sequence using expressions in terms of a. a, … , … , … [1] (iv) Use the sequence in part (c)(iii) to show algebraically that your product difference in part (c)(i) is correct. [3] Question 3 is printed on the next page. * 0000800000007 * ,










, ĬÓú¾Ġ´íÈõÏĪÅĊÞû¶þ× Ĭą²ùÙĥćđéýöô¿ÂĈăĉĂ ĥõąĕµÕĥµĕåĥÅÅÕåÕąÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN

8 0607/51/M/J/25 © UCLES 2025 3 Find the sequence that has a product difference of 578 and a last term of 100. … , … , … , 100 [3] 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. * 0000800000008 * ,










, ĬÑú¾Ġ´íÈõÏĪÅĊàù¶Ā× Ĭą²üÙğõĘÎăíûĝĞæēñĂ ĥÅÕĕõÕĥĕõąĕÅąÕąÕĕÕ DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN DO NOT WRITE IN THIS MARGIN

This document consists of 9 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) May/June 2025 MARK SCHEME Maximum Mark: 40 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.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 2 of 9 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.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 3 of 9 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.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 4 of 9 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 5 of 9 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 6 of 9 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 7 of 9 Question Answer Marks Partial Marks 1(a)(i) 7 8 2 B1 for each 1(a)(ii) Sequence Inside pair Outside pair Product difference 1, 2, 3, 4 2  3 = 6 1  4 = 4 6 – 4 = 2 10, 11, 12, 13 11  12 = 132 10  13 = 130 132 – 130 = 2 16, 17, 18, 19 17  18 = 306 16  19 = 304 306 – 304 = 2 24, 25, 26, 27 25  26 = 650 24  27 = 648 650 – 648 = 2 98, 99, 100, 101 99  100 = 9900 98  101 = 9898 9900 – 9898 = 2 6 B1 for each 2 correct cells 1(b)(i) 500  501 = 250 500 499  502 = 250 498 250 500 – 250 498 = 2 2 B1 for first two lines B1 for last line 1(b)(ii) 3.5  4.5 and 2.5  5.5 or 15.75 and 13.75 C1 FT their 4.5 and their 5.5 on both lines 4.5, 5.5 1 (product difference) 2 1 FT their 4.5 and their 5.5 1(c)(i) 4, 2 1 1(c)(ii) (a + 1)(a + 2) and a(a + 3) or correct explicit substitution 1 a2 + 2a + a + 2 a2 + 3a 2 B1 for each expansion Correct working leading to 2 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 8 of 9 Question Answer Marks Partial Marks 2(a)(i) Sequence Increase Inside pair Outside pair Product difference 1, 3, 5, 7 2 15 7 8 6, 8, 10, 12 2 80 72 8 7.2, 10.2, 13.2, 16.2 3 134.64 116.64 18 20, 24, 28, 32 4 672 640 32 88, 92, 96, 100 4 8832 8800 32 4 B1 for each row One calculation to justify the required terms of a sequence or one multiplication of a correct pair or subtraction of (their) inner and (their) outer pairs or in row three: 134.64 – 18 = 116.64 C1 2(a)(ii) 6000 or 1 correct trial with difference of 5 ≠ 6000 or 1 correct trial without difference of 5 = 6000 or sketch of x(x + 5) C1 75 × 80 = 6000 or correct sketch of x(x + 5) = 6000 C1 70, 75, 80, 85 1 2(b)(i) 72 on row 6 1 Three of the differences 6, 10, 14, 18, 22 seen or calculation of correct product difference for their 4-term sequence with increase 6 C1 2(b)(ii) 2n2 oe 1 Three common differences of 4 seen or Square numbers 1, 4, 9, [16] seen or Three of × 2, × 4, × 6, × 8, × 10, × 12 or 2×, 4×, 6×, 8×, 10×, 12× seen in table C1 2(c)(i) 242 1 2  112 C1 FT their algebraic expression in n 2(c)(ii) 23, 34, 45 1 2(c)(iii) a + 11, a + 22, a + 33 oe 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 9 of 9 Question Answer Marks Partial Marks 2(c)(iv) ( )( ) 2 11 22 33 242 a a a a + + = + + 1 FT their two term expressions in 2(c)(iii) ( ) 2 33 33 a a a a + = + 1 242 with correct working 1 3 Sketch of 2n2 or table of values or 578 2 oe or two trials of their expression including one using 17 or continuation of the sequence of product differences 72, 98, 128, 162, 200, 242, 288, 338, 392, 450, 512, [578] C1 FT their 2n2 17 seen 1 49, 66, 83 1 FT their 17 if correctly calculated