Cambridge IGCSE Mathematics - International 0607 — 2025 Oct/Nov Paper 6 · Variant 1

This document has 12 pages. [Turn over Cambridge IGCSE™ DC (CE/JG) 345051/2 © UCLES 2025 * 4 9 7 3 7 9 9 4 7 3 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 Investigation and Modelling (Extended) October/November 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 [ ]. ,










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2 0607/61/O/N/25 © UCLES 2025 Section A INVESTIGATION SQUARE RINGS You are advised to spend no more than 45 minutes on this section. 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 part of this number grid. 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. 80 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 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 * 0000800000002 * ,










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3 0607/61/O/N/25 © UCLES 2025 [Turn over 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] (b) Complete this table using part (a) and any patterns you notice. Square ring (n) L S Ring value L - S 1 2 222 74 148 3 4 246 82 164 5 86 [2] (c) Find an expression, in terms of n, for the ring value. … [3] (d) Find the ring value for square ring 731. … [2] * 0000800000003 * ,










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4 0607/61/O/N/25 © UCLES 2025 2 6 by 6 square rings On the grid, a 6 by 6 square surrounds a 4 by 4 square. The square ring has width 6. 80 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 90 81 82 83 84 85 86 87 88 89 (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 1 5 2 13 14 15 2 35 45 44 43 42 32 22 = + + + + + + + + + + + = … Ring value = … [1] (b) Complete the table. Square ring (n) L S Ring value L - S 1 570 2 590 236 3 610 366 4 630 378 5 650 390 260 [2] * 0000800000004 * ,










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5 0607/61/O/N/25 © UCLES 2025 [Turn over (c) Find an expression, in terms of n, for the ring value. … [2] (d) A square ring has a ring value of 1028. Find the number in the bottom right square of this square ring. … [3] * 0000800000005 * ,










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6 0607/61/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 8n + kw - k. For an 8 by 8 square ring, an expression for the ring value is 8n + 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 0607/61/O/N/25 © UCLES 2025 [Turn over 4 The diagram shows two square rings, shaded and . The larger square ring has width w. The number in the top left square is n. w w n Use algebra to find what is always true about the difference between the two ring values. … [5] * 0000800000007 * ,










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8 0607/61/O/N/25 © UCLES 2025 Section B MODELLING LIFT You are advised to spend no more than 45 minutes on this section. In this task, you will model the upward force, lift, that an aircraft wing makes when it flies through the air. Different wing shapes make different amounts of lift. This task is about one wing shape and the lift that it makes. wing lift airflow Direction of flight 5 We need to know the air density (the mass of 1 cubic metre of air) to calculate the lift. This table shows the air density at different heights above sea level. Height above sea level (h km) 0 1 5 10 20 32 40 47 50 Air density (d kg/m3) 1.2250 1.1160 0.7361 0.4127 0.0880 0.0132 0.0039 0.0014 0.0010 The values in the shaded cells are plotted on the grid. 0 0.4 0.6 0.8 1.0 1.2 Air density (kg/m3) Height above sea level (km) d h 0 10 20 30 40 50 0.2 * 0000800000008 * ,










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9 0607/61/O/N/25 © UCLES 2025 [Turn over (a) Plot the remaining two points on the grid. [1] (b) Model A A possible model for air density is d h 39 160 195 = + for . h 0 50 G G (i) Sketch this model on the grid. Label the sketch A. [2] (ii) Estimate the range of values for h when the model will give (a) a value below the actual air density … [1] (b) a value above the actual air density. … [1] (c) Model B Another possible model is ( ) . log d h 2 3 1 0 5 = + - for . h 0 50 G G (i) Sketch this model on the grid. Label the sketch B. [2] (ii) (a) Estimate the range of values for h when the model is valid. … [1] (b) Comment on the model when h = 50. … [1] * 0000800000009 * ,










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10 0607/61/O/N/25 © UCLES 2025 6 To make lift, an aircraft wing is angled to the airflow. This is called the angle of attack. lift angle of attack airflow As the speed of the aircraft decreases, the angle of attack, a°, increases to get more lift. The amount of lift depends on a quantity known as the lift coefficient, C. This table shows how C changes for different values of the angle of attack. Angle of attack (a° ) -5 0 5 10 15 20 25 Lift coefficient (C ) 0.00 0.55 1.05 1.46 1.70 1.66 1.44 A possible model for C is ( ) C m n a 17 2 = + - where m and n are constants. (a) Use a = 0 and a = 20 to find two equations each in terms of m and n. Give each equation in its simplest form. … … [3] * 0000800000010 * ,










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11 0607/61/O/N/25 © UCLES 2025 [Turn over (b) Find the value of m and the value of n, and write the model for C. Give each value correct to 2 significant figures. m = … n = … C = … [4] (c) Use your model to find the maximum lift coefficient for this wing. … [2] Question 7 is printed on the next page. * 0000800000011 * ,









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12 0607/61/O/N/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. 7 A model for the lift, L newtons, that an aircraft makes is L dv AC 2 2 = . d = air density ( ) kg/m3 v = speed (m/s) A = wing area ( ) m2 C = lift coefficient An aircraft has a wing area of m 125 2. It flies • at a height of 12 km above sea level • at a speed of 210 m/s • with angle of attack 3°. Use Question 6(b) and Model A in Question 5(b) to calculate the lift for this aircraft. Give your answer correct to 2 significant figures. … [7] * 0000800000012 * ,









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This document consists of 10 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 2025 MARK SCHEME Maximum Mark: 60 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.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 2 of 10 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 October/November 2025 © Cambridge University Press & Assessment 2025 Page 3 of 10 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 October/November 2025 © Cambridge University Press & Assessment 2025 Page 4 of 10 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 October/November 2025 © Cambridge University Press & Assessment 2025 Page 5 of 10 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 October/November 2025 © Cambridge University Press & Assessment 2025 Page 6 of 10 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 either their 234 or their 78 (but not both) 1(b) (n) L S L – S 1 210 70 140 2 222 74 148 3 234 78 156 4 246 82 164 5 258 86 172 2 B1 for 2 correct cells

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 7 of 10 Question Answer Marks Partial Marks 1(c) 8n + 132 oe 2 B1 for 8n + k or jn + 132, j  0 Three differences of 8 in final column or in working C1 1(d) 5980 2 M1 for substitution of n = 731 in their (8n + 132) or B1 for 8970 – k or h – 2990, k and h ≠ 0 2(a) 342 228 1 2(b) (n) L S L – S 1 570 342 228 2 590 354 236 3 610 366 244 4 630 378 252 5 650 390 260 2 B1 for 2 correct cells 2(c) 8n + 220 oe 2 B1 for 8n + k or jn + 220, j  0 2(d) 8n + 220 = 1028 C1 FT their(8n + 220) 156 2 B1 for 101 or for correct solution to their equation provided their(8n + 220) is of, or would reduce to, the form an + b (a ≠ 1, b ≠ 0) OR B1 for 55 more than their positive integer for 101 3 8k – k = 308 or 6k – k = 220 or 4k – k = 132 or C1 (k =) 44 8n + 44w – 44 oe 2 B1 for each

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 8 of 10 Question Answer Marks Partial Marks 4 [top left small] n + 11 or n + 10 + 1 and [width small] w – 2 C2 C1 for [top left small] n + 11 or [width small] w – 2 8(n + 11) + 44(w – 2) – 44 followed by correct expansion of either 8(n + 11) or 44(w – 2) C2 For C1 or C2 FT 8n + (their44)w – their44 C1 for correct expansion of either 8(n + 11) or (their44)(w – 2) Always 0 oe 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 9 of 10 Question Answer Marks Partial Marks Modelling LIFT 5(a) Two “plots” in approx correct vertical position on h = 10 and h = 20 1 5(b)(i) Correct sketch 2 B1 for correct shape always above h axis reaching h = 45 B1 for curve with d intercept at approx. 1.2 5(b)(ii)(a) h < k where 10 ≤ k < 20 1 5(b)(ii)(b) h > m where their k ⩽ m ⩽ 20 1 FT their positive k 5(c)(i) Correct sketch 2 B1 for correct shape intersecting h-axis between 40 and 50 B1 for curve with intention to meet d - axis above 1.3 5(c)(ii)(a) 15 to 20 < h < 40 to 49 1 5(c)(ii)(b) negative (so) invalid oe 1 6(a) 0.55 = m + 289n and 1.66 = m + 9n 3 B2 for one correct or B2 for 0.55 = m + n(0 – 17)2 and 1.66 = m + n(20 – 17)2 or for C = m + 289n and C = m + 9n or B1 for 0.55 = m + n(0 – 17)2 or 1.66 = m + n(20 – 17)2 or for C = m + 289n or C = m + 9n

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 10 of 10 Question Answer Marks Partial Marks 6(b) [m =] 1.7 [n =] – 0.0040 [ 2 1.7 0.0040( 17) = − − C a ] 2 B1 for each or for m = 1.69…. and n = – 0.00396... C marks are only available for equations containing just 2 variables If candidate restarts with different a and C values award C marks for correct method Correct method to eliminate one variable or Correct sketch showing two intersecting lines with appropriate gradients C1 FT their final equations in m and n in 6(a) Correct method to find second variable or Correct sketch with point of intersection labelled C1 FT their final equations in m and n in 6(a) 6(c) [a =] 17 seen C1 1.70 1 FT their positive numerical m from (b) 7 Correct substitution of a = 3 in model in 6(b) C1 FT their model for C in terms of a only [C =] 0.916 to 0.923[0...] seen 1   195 39 12 160 =  + d C1 [d =] 0.311 or 0.3105... seen 1 Correct substitution of their d and their C and 210 and 125 in 2 2 = dv AC L C1 780 000 or 790 000 2 B1 for 783 000 ⩽ answer < 792 000