Cambridge IGCSE Mathematics - International 0607 — 2025 May/June Paper 5 · Variant 3

This document has 12 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 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 [ ]. * 3 3 2 5 7 7 0 2 0 3 * DC (PQ/CB) 346985/3 © UCLES 2025 ,










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2 0607/53/M/J/25 © UCLES 2025 INVESTIGATION NUMBER PYRAMIDS In this investigation you will look at the numbers in a pyramid of numbers. The numbers are placed in order in rows. Each row has one more number than the row above. 1 2 3 37 Row 9 38 40 41 29 30 31 32 33 34 35 22 23 24 25 26 27 16 17 18 19 20 11 12 13 14 7 8 9 4 5 6 10 15 21 28 36 45 Row 8 Row 7 Row 6 Row 5 Row 4 Row 3 Row 2 Row 1 1 (a) Complete row 9 in the pyramid. [1] (b) In row 4 the 2nd number is 8. We write ( , ) 4 2 8 N = . In row 4 the 3rd number is 9. We write ( , ) 4 3 9 N = . In row 4 the last number is 10. We write ( , ) 4 4 10 N = . (i) Complete the statement. In row 3 the 1st number is 4. We write N (… , …) = 4. [1] (ii) Complete the statements. N (6, 3) = … N (5, …) = 12 N (…, 4) = 25 [2] (c) Complete the statement. In row 8 the last number is written N (… , …). [1] * 0000800000002 * ,










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3 0607/53/M/J/25 © UCLES 2025 [Turn over 2 The numbers at the end of the rows make this sequence. 1, 3, 6, 10, 15, ... (a) Complete the statement. This is the sequence of … numbers. [1] (b) (i) Complete the table. Use the pyramid of numbers and patterns to help you. Row number Calculation Last number in the row 1 1 # 2 = 2 1 2 2 # 3 = 3 3 3 # = 6 4 # = 5 # = 6 6 # 7 = 42 21 10 # = [3] (ii) Show that the last number in row 50 is 1275. [2] (iii) Find an expression for the last number in row R. … [1] * 0000800000003 * ,










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4 0607/53/M/J/25 © UCLES 2025 3 To find a number in the pyramid add its position in the row to the last number in the row above. Example N(6, 4) is in row 6. The last number in row 5 is 15. N(6, 4) is in position 4. So ( , ) 6 4 15 4 19 N = + = . 1 2 3 37 38 40 41 29 30 31 32 33 34 35 22 23 24 25 26 27 16 17 18 19 20 11 12 13 14 7 8 9 4 5 6 10 15 21 28 36 45 Row 6 Row 5 (a) Complete the statements. (i) N (7, 4) = … + 4 = 25 [1] (ii) N (… , …) = 45 + … = 50 [1] (b) This is a formula to find any number in row R. ( , ) ( ) R k R R k 2 1 N = - + k is the position of the number in row R. (i) Show that the formula is correct for N(6, 4). [1] * 0000800000004 * ,










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5 0607/53/M/J/25 © UCLES 2025 [Turn over (ii) Find the 10th number in row 55. … [2] (c) (i) Calculate N(26, 26). … [2] (ii) ( , ) R k 362 N = Find the value of R and the value of k. R = … k = … [3] * 0000800000005 * ,










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6 0607/53/M/J/25 © UCLES 2025 4 Numbers in the pyramid can be joined to make a diamond as shown. All diamonds in this investigation are similar. 1 2 3 37 38 40 41 29 30 31 32 33 34 35 22 23 24 25 26 27 16 17 18 19 20 11 12 13 14 7 8 9 4 5 6 10 15 21 28 36 45 1 2 3 37 38 40 41 29 30 31 32 33 34 35 22 23 24 25 26 27 16 17 18 19 20 11 12 13 14 7 8 9 4 5 6 10 15 21 28 36 45 1 2 3 37 38 40 41 29 30 31 32 33 34 35 22 23 24 25 26 27 16 17 18 19 20 11 12 13 14 7 8 9 4 5 6 10 15 21 28 36 45 2, 4, 5 and 8 make a diamond. The four numbers are always written in order from smallest to largest. This diamond has width 1 because 5 4 1 - = . We write D(2, 4, 5, 8) has width 1. 6, 13, 15 and 26 make a diamond. This diamond has width 2 because 15 13 2 - = . We write D(6, 13, 15, 26) has width 2. 1, 11, 15 and 41 make a diamond. D(1, 11, 15, 41) has width 4 because 15 11 4 - = . * 0000800000006 * ,










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7 0607/53/M/J/25 © UCLES 2025 [Turn over (a) 1 2 3 37 38 40 41 29 30 31 32 33 34 35 22 23 24 25 26 27 16 17 18 19 20 11 12 13 14 7 8 9 4 5 6 10 15 21 28 36 45 (i) Complete the statements. D( 5 , 8 , … , 13 ) has width … D(… , 18 , 20 , …) has width … D( 2 , … , … , …) has width 4. [6] * 0000800000007 * ,










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8 0607/53/M/J/25 © UCLES 2025 (ii) Violetta draws diamond D(a, b, c, 33). Her diamond has the largest possible width. Find the width and the values of a, b and c. 1 2 3 37 38 40 41 29 30 31 32 33 34 35 22 23 24 25 26 27 16 17 18 19 20 11 12 13 14 7 8 9 4 5 6 10 15 21 28 36 45 width = … a = … b = … c = … [3] (iii) For D(a, b, c, d ) write an expression for the width, w, in terms of b and c. … [1] * 0000800000008 * ,










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9 0607/53/M/J/25 © UCLES 2025 [Turn over (b) (i) Complete the table. D(a, b, c, d ) ( ) ( ) a d b c + - + Width w D(5, 8, 9, 13) ( ) 5 13 + - ( ) 8 9 + = 1 1 D(4, 11, 13, 24) ( ) 4 24 + - ( ) 11 13 + = 4 2 D(6, 18, 21, 42) ( ) - ( ) = 9 3 D(3, 17, 21, 51) ( ) - ( ) = [3] (ii) Complete this statement using an expression in terms of w. ( ) ( ) a d b c + - + = … [1] * 0000800000009 * ,










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10 0607/53/M/J/25 © UCLES 2025 (c) D(a, b, 147, 182) has w 2 = . Find the value of a and the value of b. a = … b = … [4] * 0000800000010 * ,










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11 0607/53/M/J/25 © UCLES 2025 BLANK PAGE * 0000800000011 * ,









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12 0607/53/M/J/25 © UCLES 2025 BLANK PAGE 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 9 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 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.

0607/53 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.

0607/53 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.

0607/53 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

0607/53 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

0607/53 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

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 7 of 9 Investigation: Number Pyramids Question Answer Marks Partial Marks 1(a) 37 38 39 40 41 42 43 44 45 1 1(b)(i) 3, 1 1 1(b)(ii) 18 2 7 2 B1 for 2 correct answers 1(c) 8, 8 1 2(a) triangle 1 2(b)(i) Row number Calculation Last number 1 1 × 2 = 2 1 2 2 × 3 = 6 3 3 3 × 4 = 12 6 4 4 × 5 = 20 10 5 5 × 6 = 30 15 6 6 × 7 = 42 21 10 10 × 11 = 110 55 3 B2 for rows 2 to 5 correct or B1 for one of row 4 or 5 correct and B1 for row 10 2(b)(ii) 50 51 2  2 B1 for 50 51  or their product divided by 2 If 0 scored, SC1 for showing addition of all the last numbers to get to 1275 2(b)(iii) ( ) 1 2 R R  + oe 1 3(a)(i) 21 1 3(a)(ii) N(10 , 5) = 45 + 5 = 50 1 3(b)(i) ( ) 6 6 1 4 19 2 − + = 1

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 8 of 9 Question Answer Marks Partial Marks 3(b)(ii) ( ) 55 55 1 10 2  − + C1 1495 1 3(c)(i) ( ) 26 26 1 2 − + 26 C1 351 1 3(c)(ii) 362 35 − 1 or ( ) 27 27 1 362 2 k − + = C1 FT their 351 if less than 362 27 R = 1 11 k = 1 FT their 351 if less than 362 4(a)(i) D ( 5 , 8 , 9 , 13 ) width 1 D ( 9 , 18 , 20 , 33 ) width 2 D ( 2 , 16 , 20 , 50 ) 5 B1 for row 1 B2 for row 2 or B1 for 2 correct values in row 2 B2 for row 3 or B1 for 2 correct values in row 3 For one complete diamond correctly drawn on the pyramid C1 4(a)(ii) [width =] 3 [a =] 3 [b =] 12 [c =] 15 2 B1 for either a = 3 or width = 3 If 0 scored, SC1 for 3, 12, 15 incorrectly placed For the complete diamond correctly drawn on the pyramid C1 4(a)(iii) c b − 1

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 9 of 9 Question Answer Marks Partial Marks 4(b)(i) ( ) , , , D a b c d ( ) ( ) + − + a d b c w ( ) 5 , 8 , 9 ,1 3 D ( ) ( ) 5 13 8 9 1 + − + = 1 ( ) 4 ,1 1 ,1 3 , 24 D (4+24) ( ) 11 13 4 − + = 2 ( ) 6 ,1 8 , 21 , 42 D ( ) ( ) 6 42 18 21 + − + = 9 3 ( ) 3,1 7, 21, 51 D ( ) ( ) 3 51 17 21 16 + − + = 4 3 B1 for correct third row B1 for brackets correct in fourth row B1 for 16 and 4 4(b)(ii) 2 w oe 1 4(c) 147 2 − [= 145] C1 (a + 182) – (b + 147) = 4 C1   1 14 a =   1 45 b = 2 B1 for each