Cambridge IGCSE Mathematics - International 0607 — 2025 Oct/Nov Paper 6 · Variant 3

[Turn over Cambridge IGCSE™ This document has 12 pages. Any blank pages are indicated. DC (WW/FC) 349992/4 © UCLES 2025 * 8 2 8 7 6 9 7 7 1 0 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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 0607/63/O/N/25 © UCLES 2025 Section A INVESTIGATION THE VERTEX OF THE QUADRATIC FUNCTION You are advised to spend no more than 45 minutes on this section. In this task, you will investigate the change in the vertex of the quadratic function y ax bx c 2 = + + as a or b changes. 1 In this question, the value of a changes. The value of b and the value of c stay the same. (a) y 6 4 2 0 – 2 – 4 – 6 2 4 6 (1, 1) (2, –2) x The diagram shows the graphs of these functions. . y x x 1 5 6 4 2 = - + y x x 3 6 4 2 = - + These functions are quadratic functions of the form y ax bx c 2 = + + where b 6 =- and c 4 = . The value of a changes. The coordinates of the minimum points of the quadratic functions are shown on the diagram. These points are the vertices of the quadratic functions. * 0000800000002 * ,










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3 0607/63/O/N/25 © UCLES 2025 [Turn over (i) On the diagram, sketch the graph of y x x 6 4 2 = - + . [2] (ii) The line of vertices is the line that joins the vertices of the quadratic graphs. Find the equation of the line of vertices. … [3] (iii) Does the vertex of . y x x 0 5 6 4 2 = - + lie on the line in part (ii)? … [3] * 0000800000003 * ,










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4 0607/63/O/N/25 © UCLES 2025 (b) y x 6 4 2 0 – 2 – 4 – 6 – 2 – 4 – 6 – 8 2 4 6 The diagram shows the graphs of these functions. . y x x 0 5 4 5 2 = + + y x x 4 5 2 = + + y x x 2 4 5 2 = + + These functions are quadratic functions of the form y ax bx c 2 = + + where b 4 = and c 5 = . The value of a changes. Find the equation of the line of vertices. … [2] * 0000800000004 * ,










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5 0607/63/O/N/25 © UCLES 2025 [Turn over (c) (i) Complete the table. You may use any patterns you notice. Quadratic function Equation of the line of vertices part (a)(ii) y ax x 6 4 2 = - + part (b) y ax x 4 5 2 = + + y ax x 8 20 2 = - + y x 4 20 =- + y ax x 12 30 2 = - + y ax2 = … y x 5 2 = - [3] (ii) In quadratic functions of the form y ax bx c 2 = + + , the value of b and the value of c stay the same. The value of a changes. Complete the equation of the line of vertices. y = … x + … [2] * 0000800000005 * ,










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6 0607/63/O/N/25 © UCLES 2025 2 In this question, the value of b changes. The value of a and the value of c stay the same. (a) y x 6 8 4 2 0 – 2 – 4 – 6 – 8 – 2 – 4 – 6 – 8 2 4 6 – 10 The diagram shows the graphs of these functions. y x x 4 2 = + y x x 2 2 = + y x2 = y x x 2 2 = - y x x 4 2 = - These functions are quadratic functions of the form y ax bx c 2 = + + where a 1 = and c 0 = . The value of b changes. (i) On the diagram, sketch the graphs of y x x 6 2 = + and y x x 6 2 = - . [3] (ii) Draw a smooth curve through the vertices of the 7 graphs. [1] (iii) Find the equation of the smooth curve through the vertices. … [2] * 0000800000006 * ,










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7 0607/63/O/N/25 © UCLES 2025 [Turn over (b) In this part, quadratic functions of the form y ax bx c 2 = + + have a 2 = and c 0 = . The value of b changes. Find the equation of the curve through the vertices. You may use sketches on your calculator to help you. … [2] (c) In this part, quadratic functions of the form y ax bx c 2 = + + have constant non-zero values for a and c. The value of b changes. Find the equation of the curve through the vertices. … [2] * 0000800000007 * ,










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8 0607/63/O/N/25 © UCLES 2025 Section B MODELLING CLUB FEES You are advised to spend no more than 45 minutes on this section. In this task, you will look at the relationship between the number of members of a club and the fee for each member. The fee for each member is the amount of money that each member pays to the club. 3 A club has 200 members. The fee for each member is $100. (a) Show that the total fees that all the members pay to the club are $20 000. [1] (b) 40 members leave the club. (i) The club wants to increase the fee for each member by $20. Find the change in total fees. … [3] (ii) The club decides to increase the fee for each member so that the total fees are still $20 000. Calculate the new fee for each member. … [2] * 0000800000008 * ,










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9 0607/63/O/N/25 © UCLES 2025 [Turn over 4 A club has 200 members. The fee for each member is $100. x members leave the club. The club wants the total fees to be $20 000. (a) Write down an expression for the new fee that each member pays. … [1] (b) The increase in the fee for each member is $y. Show that a model for the increase in the fee is y x x 200 100 = - . [3] (c) Write down the values of x for which the model is valid. … [1] * 0000800000009 * ,










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10 0607/63/O/N/25 © UCLES 2025 (d) Sketch the graph of the model y x x 200 100 = - for values of x between 0 and 100. x y Number of members that leave Fee increase ($) 100 0 100 0 [2] (e) Some members leave the club. The club uses the model y x x 200 100 = - and finds that the fee increase is $13. Find the number of members that left the club. … [2] * 0000800000010 * ,










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11 0607/63/O/N/25 © UCLES 2025 5 Another club has N members. The fee for each member is $F. x members leave the club. The club increases the fee for each member by $y. (a) (i) Write down an expression for the total fees before the increase in the fee. … [1] (ii) Complete the expression for the total fees after the increase in the fee. (… - …)(… + …) [2] (b) The total fees in part (a)(i) and part (a)(ii) are the same. Find a model for x in terms of N, F and y. Write your answer as a single fraction. … [4] (c) 20 members leave the club. The club increases the fee for each member by $10 so that the total fees in part (a)(i) stay the same. Find a model for N in terms of F. … [3] * 0000800000011 * ,









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12 0607/63/O/N/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 10 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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.

0607/63 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.

0607/63 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.

0607/63 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

0607/63 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

0607/63 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)(i) Correct sketch passing through (3, –5), (0, 4) and approx.. (5.5, 1) 2 B1 for correct shape 1(a)(ii) Line of vertices drawn with triangle marked and sides given or gradient formula with correct substitution C1 y = –3x + 4 2 B1 for gradient = –3 or B1 for y = mx + 4 1(a)(iii) x [coordinate of vertex] = 6 oe C1 y = –3×6 + 4 C1 FT their –3x + 4 Yes and –14 seen twice 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 7 of 10 Question Answer Marks Partial Marks 1(b) y = 2x + 5 2 B1 for 2x + 5 or y = 2x + k or y = mx + 5 m ≠ 0 1(c)(i) Function Equation y = ax2 – 12x + 30 y = – 6x + 30 y = ax2 + 10x – 2. y = 5x – 2 3 B1 for y =, + 30 and – 2 B1 for – 6x B1 for + 10x 1(c)(ii) 2 b oe c 2 B1 for each 2(a)(i) Correct sketch of parabolas 3 B1 for two U shaped curves B1 for symmetrical diagram about y-axis B1 for curve reaching (0, 0) with vertex at (3, –9) or (–3, –9) 2(a)(ii) Smooth curve through at least the given vertices 1 2(a)(iii) At least 3 correct vertices listed oe C1 y = –x2 oe 1 2(b) At least 3 correct vertices listed or at least 3 sketches of y = 2x2 + bx and the curve through their vertices. C1 y = –2x2 oe 1 2(c) y = – ax2 + c oe 2 B1 for –ax2 seen

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 8 of 10 Question Answer Marks Partial Marks Modelling Club Fees 3(a) 200 × 100 1 3(b)(i) 160×120 or 40×100 and 160 × 20 C1 20 000 – 19 200 or 19 200 – 20 000 or 4000 – 3200 or 3200 – 4000 C1 FT their 19 200 FT their 3200 $800 [less] or [–] $800 oe 1 FT their subtraction from 20 000 or 4000 3(b)(ii) 20 000 ÷ 160 or 120 + 800 160 or 100 + 40 100 160  C1 FT their 160 FT their 800 $125 1 FT their 160 if < 200 4(a) 20000 200−x or 100 100 200 + − x x isw 1 4(b) 20000 200−x – 100 1 FT their 20000 200−x Correct next step e.g 20000 100(200 ) 200 200 − − − − x x x 1 leading to y = 100 200− x x 1 Scheme for when answer to 4(a) is 100 100 200 + − x x 100 100 200 + − x x – 100 2 B1 for 100 100 200 + − x x = 100 + y leading to y = 100 200− x x 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 9 of 10 Question Answer Marks Partial Marks 4(c) [0 ⩽ x ] < 200 [where x is an integer] 1 4(d) Correct shape through the origin. 1 Line must have continuously increasing positive gradient Reaches (100, 100) C1 Dependent on correct shape 4(e) Horizontal line y = 13 on diagram or 100 200− x x = 13 or (200 – x)(100 + 13) = 20000 C1 23 1 5(a)(i) NF 1 5(a)(ii) (N – x)(F + y) or (N – x)(y + F) isw 2 B1 for one correct bracket 5(b) [(N – x)(F + y) = ] NF + Ny – xF – xy or x = N – + NF F y or   =   −   Fx y N x y(N – x) = yN – yx C2 FT their algebraic binomial brackets C1 for three terms correct or C1 for N – x = + NF F y or C1 for y(N – x) = Fx or Fx = yN – yx x = + Ny F y 2 B1FT for isolating x terms on one side of their equation e.g. xF + xy = Ny or B1FT for common denominator ( ) + − + + N F y NF F y F y or better

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2025 © Cambridge University Press & Assessment 2025 Page 10 of 10 Question Answer Marks Partial Marks 5(c) 20 = 10 10 + N F or 10 = 20 20 − F N or FN = (N – 20)(F + 10) C1 FT substitution of y = 10 and x = 20 in their model or in   =   −   Fx y N x oe or in FN = (N – x)(F + y) N = 2F + 20 or N = 2(F + 10) oe 2 FT their 10 20 10 N F   =   +   or their 20 10 20 F N   =   −   if of the form 20 = + + aN b cF d or 10 = + + aF b cN d (b may be 0) B1 for correct first step e.g. 20F + 200 = 10N or 10N – 200 = 20F or 2 = [1] 10 + N F or 1 = 2 20 − F N or [FN =] NF + 10N –20F – 200