Cambridge IGCSE Mathematics - International 0607 — 2025 Oct/Nov Paper 5 · Variant 1

This document has 8 pages. [Turn over * 7 4 5 7 6 4 5 6 4 6 * Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 Investigation (Core) October/November 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/CGW) 345048/2 © UCLES 2025 ,










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2 0607/51/O/N/25 © UCLES 2025 INVESTIGATION SQUARE RINGS In this investigation, you will work with the numbers along the inside edges of two squares drawn on a number grid of width 10. The diagram shows a 10 by 10 grid with the numbers 1 to 100. On the grid, a 4 by 4 square surrounds a 2 by 2 square. The shaded area forms a 4 by 4 square ring. A square ring is always one square thick. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 The ring value is worked out in this way. step 1 Add the numbers along the inside edges of the large square to get L. step 2 Add the numbers along the inside edges of the small square to get S. step 3 Calculate L S - . This is the ring value. Example step 1 L = 3 4 5 6 16 26 36 35 34 33 23 13 + + + + + + + + + + + step 2 S = 14 15 25 24 + + + step 3 Ring value = L S - 1 4 by 4 square rings The example shows square ring 3. This is because 3 is the top left number in the ring. (a) Work out the ring value for the square ring in the example. … [2] * 0000800000002 * ,










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3 0607/51/O/N/25 © UCLES 2025 [Turn over (b) Work out the ring value for square ring 1, which has 1 in the top left corner. You may use this grid to help you. … [3] (c) Complete this table using part (a), part (b) and any patterns you notice. Square ring (n) L S Ring value L S - part (b) 1 2 222 74 148 part (a) 3 4 246 164 5 86 [3] (d) An expression for the ring value is gn 132 + . Find the value of g and write down the expression. … [2] (e) Mina draws a 4 by 4 square ring on the 10 by 10 grid. Mina’s square ring has the largest possible ring value. Find the ring value. … [3] * 0000800000003 * ,










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4 0607/51/O/N/25 © UCLES 2025 2 6 by 6 square rings The grid continues downwards beyond the number 100. A 6 by 6 square surrounds a 4 by 4 square. The square ring has width 6. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 (a) Complete the working to find the ring value for square ring 1. L = 1 2 3 4 5 6 16 26 36 46 56 55 54 53 52 51 41 31 21 11 + + + + + + + + + + + + + + + + + + + = 570 S = 12 13 14 15 25 35 45 44 43 42 32 22 + + + + + + + + + + + = … Ring value = … [2] (b) Complete the table. Square ring (n) L S Ring value L S - 1 570 2 590 236 3 610 366 4 630 5 650 260 [3] * 0000800000004 * ,










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5 0607/51/O/N/25 © UCLES 2025 [Turn over (c) Find an expression, in terms of n, for the ring value. … [3] (d) Explain why moving the square ring one square to the right increases the values in column L by 20. … … [1] (e) Find the ring value for square ring 45. … [2] (f) A square ring has a ring value of 1028. Find the number in the bottom right square of this square ring. … [4] * 0000800000005 * ,










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6 0607/51/O/N/25 © UCLES 2025 3 w by w square rings The width of a square ring is w. For the square ring in Question 1, w 4 = . For the square ring in Question 2, w 6 = . For a w by w square ring, an expression for the ring value is n kw k 8 + - . For an 8 by 8 square ring, an expression for the ring value is n 8 308 + . Find the value of k. Write down an expression, in terms of n and w, for the ring value of a w by w square ring. k = … expression for the ring value … [3] * 0000800000006 * ,










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7 0607/51/O/N/25 © UCLES 2025 [Turn over 4 (a) Square ring 71 has a ring value of 832. Find the width of this square ring. … [4] Question 4(b) is printed on the next page. * 0000800000007 * ,










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8 0607/51/O/N/25 © UCLES 2025 (b) In the diagram, a 6 by 6 square ring surrounds a 4 by 4 square ring. The 4 by 4 square ring has a ring value of 1300. Find the difference between the ring values for these two square rings. … [5] 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 * ,










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This document consists of 9 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) October/November 2025 MARK SCHEME Maximum Mark: 36 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 October/November 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 October/November 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 October/November 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 October/November 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 October/November 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 October/November 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 Question Answer Marks Partial Marks 1(a) 234 – 78 C1 156 1 FT their 234 or their 78 (but not both) 1(b) 1 + 2 + 3 + 4 + 14 + 24 + 34 + 33 + 32 + 31 + 21 + 11 and 12 + 13 + 23 + 22 or correct grid completely filled or identified on first grid C1 140 2 B1 for [L =] 210 or [S =] 70

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 7 of 9 Question Answer Marks Partial Marks 1(c) Square ring (n) L S Ring value L – S 1 210 70 140 2 222 74 148 3 234 78 156 4 246 82 164 5 258 86 172 3 B1 for each correct cell 1(d) 8n + 132 1 Substitution for n and corresponding value of L – S. e.g. 2g +132 = 148 or 148 1 32 2 − or three differences of 8 in final column or in workspace C1 1(e) 668 2 B1 for 67 identified Correct substitution of their 67 in their (8n + 132) or 1002 – 334 C1 2(a) 342 228 2 B1 for each FT 570 – their 342 2(b) Square ring (n) L S Ring value L – S 1 570 342 228 2 590 354 236 3 610 366 244 4 630 378 252 5 650 390 260 3 B2 for 4 cells correct or B1 for 2 cells correct 2(c) 8n + 220 oe 2 B1 for 8n + k or jn + 220 , j  0 Three differences of 8 in final column C1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 8 of 9 Question Answer Marks Partial Marks 2(d) [There are] 20 squares and [Each number] increases by 1 oe 1 2(e) Correct substitution of 45 in their (8n + 220) or 1450 – 870 C1 580 1 2(f) 8n + 220 = 1028 C1 FT their(8n + 220) 156 3 B2 for 101 or correct solution to their equation provided their(8n + 220) is of, or would reduce to, the form an + b (a ≠ 1, b ≠ 0) or B1FT for correct working to find n e.g. 1028 220 8 − or solution to their equation with a = 1 or b = 0. OR B1FT for 55 more than their positive integer 101 3 [k =] 44 8n + 44w – 44 oe 2 B1 for each 8k – k = 308 or 6k – k = 220 or 4k – k = 132 C1 4(a) n = 71 substituted in their (8n + 44w – 44) = 832 or n = 71 substituted: 8n + 220 gives 788 for 6 by 6 square 8n + 308 gives 876 for 8 by 8 square C2 C1 for n = 71 substituted in their (8n + 44w – 44) or C1 for their (8n + 44w – 44) = 832 or C1 for their 8×71 + kw – k = 832 or C1 for n = 71 substituted: 8n + 220 gives 788 for 6 by 6 square or 8n + 308 gives 876 for 8 by 8 square correct isolation of w or 832 is [halfway] between 788 and 876 C1 7 nfww 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 9 of 9 Question Answer Marks Partial Marks 4(b) 0 supported by some correct working 5 Dep on 146 as top left of inner ring B2 for 146 as top left of inner ring or M1 for their 8n + 132 = 1300 or 1300 1 32 8 − their AND B1FT for 11 less than their integer 146 seen M1 for their integer 135 correctly substituted in their (8n + 220)