Cambridge IGCSE Mathematics - International 0607 — 2025 May/June Paper 6 · Variant 2

This document has 16 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 Investigation and Modelling (Extended) May/June 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 [ ]. * 6 4 7 4 3 5 6 9 6 3 * DC (PQ/SG) 347707/3 © UCLES 2025 ,










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2 0607/62/M/J/25 © UCLES 2025 Section A INVESTIGATION THE CORE AND SHELL OF LINEAR SEQUENCES You are advised to spend no more than 45 minutes on this section. In this investigation you will look at the difference between the values of the shell and the core in linear sequences. The linear sequences have a constant positive difference between consecutive terms. Each sequence in this investigation has an odd number of terms. Each term is a positive integer. The core integer is the integer in the middle of the sequence. The shell integers are the integer at the beginning and the integer at the end of the sequence. Example For the sequence core integer shell integers 15 12 9 6 18 21 24 the core integer is 15 and the shell integers are 6 and 24. You calculate the value of the core by multiplying the core integer by itself. 15 15 225 # = You calculate the value of the shell by multiplying the shell integers together. 6 24 144 # = F is the positive difference between the value of the core and the value of the shell. F 225 144 81 = - = 1 (a) Write down the core integer and the shell integers for the sequence 5 7 9 11 13 15 17 19 21. Core integer … Shell integers … and … [1] (b) Calculate F for the sequence in part (a). … [2] * 0000800000002 * ,










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3 0607/62/M/J/25 © UCLES 2025 [Turn over 2 (a) Write down a linear sequence with 5 terms. … , … , … , … , … [1] (b) For your sequence in part (a) find the value of F. Write F as the square of a number. … [3] 3 The general linear sequence has 1st term a and common difference d. So the first 3 terms of this sequence are a, a d + , a d 2 + . (a) Write the 7th term of the general linear sequence in terms of a and d. … [1] (b) A general linear sequence has 7 terms. Find an expression for F. Write your answer in the form ( ) kd 2 where k is a positive integer. … [3] * 0000800000003 * ,










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4 0607/62/M/J/25 © UCLES 2025 4 (a) Complete the table. Linear sequence with common difference d Number of terms n F 1st term Middle term Last term a a d + a d 2 + 2 ` j a a d 2 + 5 2 ` j a 7 2 ` j a a d 4 + a d 8 + 2 ` j a a d 10 + d 5 2 ` j [4] * 0000800000004 * ,










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5 0607/62/M/J/25 © UCLES 2025 [Turn over (b) The 1st term of a linear sequence of positive integers is 1. The value of F is 36. Use the pattern in the F column in part (a) to help you find all the possible sequences. [5] * 0000800000005 * ,










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6 0607/62/M/J/25 © UCLES 2025 5 (a) A linear sequence has n terms. The common difference is d. F d 100 2 = Find the number of terms in the sequence. … [2] (b) Another linear sequence has n terms. The common difference is d. F d4 = . (i) Find n in terms of d. … [2] (ii) Give an example of a linear sequence with 11 terms where F d4 = . … [1] * 0000800000006 * ,










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7 0607/62/M/J/25 © UCLES 2025 [Turn over Section B MODELLING HAMMER THROWING You are advised to spend no more than 45 minutes on this section. In this task you will look at the sport of hammer throwing. The task uses models to find the horizontal distance that an athlete can throw the hammer. The hammer is a metal ball. The metal ball is fixed to a wire with a handle. A hammer throw has two stages. Stage 1 Stage 2 The athlete rotates the hammer holding it by the handle. The athlete releases the hammer, which travels in a parabola before hitting the ground. Athletes want to maximise the horizontal distance that the hammer travels after release. 6 Stage 1 The athlete rotates the hammer holding it by the handle. The hammer travels along a circle of radius 1.8 m as shown in this diagram. (a) The circumference, C, of a circle of radius r is C r 2r = . Calculate how far the hammer travels in one complete turn. 1.8 m … [2] (b) The hammer makes N complete turns in one second. The speed of the hammer is V m/s. Write down a model for V in terms of N. … [1] * 0000800000007 * ,










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8 0607/62/M/J/25 © UCLES 2025 7 Stage 2 2 m A° The diagram shows an athlete releasing the hammer at a speed of V m/s at an angle A to the horizontal. A is the release angle. The athlete releases the hammer at a speed of 20 m/s with release angle 30°. As the hammer travels it moves both vertically and horizontally as shown. horizontal speed vertical speed S A = 30° V = 20 This is a model for the vertical speed S m/s. sin S V A = (a) Calculate the vertical speed of the hammer. … [2] * 0000800000008 * ,










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9 0607/62/M/J/25 © UCLES 2025 [Turn over (b) In this part assume that gravity has no effect. The hammer travels vertically for x seconds at the speed that you calculated in part (a). (i) Write down an expression in terms of x for the number of metres that the hammer travels vertically. … [1] (ii) The athlete releases the hammer when it is 2 metres above the ground. Change your expression in part (b)(i) to give the number of metres that the hammer is above the ground after x seconds. … [1] (c) Gravity reduces the height in part (b)(ii) by . x 4 9 2 metres. (i) Write down a model for the height in metres, H, of the hammer above the ground after x seconds. … [1] (ii) When the hammer hits the ground, H 0 = . The hammer hits the ground after x seconds. Find the value of x. … [3] * 0000800000009 * ,










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10 0607/62/M/J/25 © UCLES 2025 8 The best release angle A is approximately 45°. This angle gives the greatest horizontal distance that the hammer travels after release. An Olympic athlete uses the best release angle. The athlete releases the hammer at a speed of V m/s when the hammer is 2 metres above the ground. The hammer hits the ground after x seconds. (a) sin S V A = and ° . sin45 0 7 = . Use Question 7(c)(i) to show that . ( . ) x V x 4 9 0 7 2 0 2 - - = . [1] * 0000800000010 * ,










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11 0607/62/M/J/25 © UCLES 2025 [Turn over (b) The formula for the solution of ax bx c 0 2 + + = is x a b b ac 2 4 2 ! = - - . (i) Use this formula to solve . ( . ) x V x 4 9 0 7 2 0 2 - - = and show that a model for x is x V V 14 80 2 = + + . [3] (ii) The solution x V V 14 80 2 = - + has been rejected because x cannot be negative. Give a reason why V V 14 80 2 - + always gives a negative result. … [1] * 0000800000011 * ,









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12 0607/62/M/J/25 © UCLES 2025 9 An athlete uses the best release angle. The hammer has a constant horizontal speed which is equal to the hammer’s vertical speed at the moment of release. (a) An athlete uses the best release angle A to throw a hammer at a speed of 20 m/s. The horizontal distance that the hammer travels is D metres. 2 m A° D metres (i) Use . sinA 0 7 = and the model sin S V A = to find the horizontal speed. … [1] (ii) Use the model in Question 8(b)(i) to find the value of D. … [3] (b) An athlete throws a hammer at V m/s using the best release angle. Use the model in Question 8(b)(i) to show that a model for D is . D V V V 20 80 2 2 = + + [1] * 0000800000012 * ,









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13 0607/62/M/J/25 © UCLES 2025 (c) In 2016 Anita Włodarczyk from Poland set the Women’s World Record for hammer throwing. Using the best release angle, she threw the hammer 82.98 metres. (i) Use the model in part (b) to find the speed V m/s at which she released the hammer. This was at the end of Stage 1 of her throw. … [2] (ii) In Question 6(b), at the end of Stage 1, the rate of turning is N turns per second. Find the value of N for Anita Włodarczyk’s record throw. … [2] * 0000800000013 * ,









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16 0607/62/M/J/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. BLANK PAGE * 0000800000016 * ,









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This document consists of 11 printed pages. © Cambridge University Press & Assessment 2025 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) May/June 2025 MARK SCHEME Maximum Mark: 50 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.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 2 of 11 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.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 3 of 11 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.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 4 of 11 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

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 5 of 11 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

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 6 of 11 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

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 7 of 11 Question Answer Marks Partial Marks Investigation: The core and shell of linear sequences 1(a) [Core integer] 13 [Shell integers] 5 and 21 1 1(b) 13 × 13 – 5 × 21 or 169 – 105 C1 FT their 13 64 1 2(a) A correct linear sequence of 5 positive integers 1 2(b) Correct working to find F C1 F written as a square 2 FT their linear sequence of 5 terms B1 for correct value of F or for their F written as a square. 3(a) a + 6d 1 3(b) ( )( ) 2 2 3 3 6 9 a d a d a ad d + + = + + C1 ( ) 2 6 6 a a d a ad + = + C1 FT their 7th term in form a kd + , k > 2 ( ) 2 3d 1 4(a) 1st Middle Last n F a a + d a + 2d 3 2 d a a + 2d a + 4d 5 ( ) 2 2d a 7 (3d)2 a a + 4d a + 8d 9 ( ) 2 4d a a + 5d a + 10d 11 (5d)2 2 B1 for 6 correct cells Correct working for F in 1st, 2nd or 4th row showing brackets and their expansions. C2 C1 for a correct expansion for the core (may be without brackets)

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 8 of 11 Question Answer Marks Partial Marks 4(b) Pairing k and d ( ) 2 36 kd = or 2 2 36 k d = or 6 kd = oe where k may be replaced by 2, 3, 4, 5 or 6 or 6 2 3 =  or ( ) 2 36 3 2 =  and correct linked pair of k and d C2 C1 for ( ) 2 36 kd = or 2 2 36 k d = or 6 kd = oe where k may be replaced by 2, 3, 4, 5 or 6 or factors of 6 or correct linked pair of k and d OR Using middle term and differences Middle term  1 36 7 a kd = + = + = and using differences to find other terms in 1, …………, 7 C1 for middle term = 7 or last term = 13 or 2 7 13 36 − = oe OR Pairing k and n or d and n 1 2 − n = k or n = 2k + 1 oe where k may be replaced by 1, 2, 3 or 6 or d = 12 1 − n or n = 12 1 + d oe where d or n may be replaced by a number and correct linked pair of k and n or d and n C1 for 1 2 − n = k or n = 2k + 1 where k may be replaced by 1, 2, 3 or 6 or d = 12 1 − n or n = 12 1 + d where d or n may be replaced by a number or for a correct linked pair of k and n or d and n 1 7 13 1 4 7 10 13 1 3 5 7 9 11 13 1 2 3 4 5 6 7 8 9 10 11 12 13 3 B2 for 3 correct, 0 or 1 wrong or 4 correct, 1 or 2 wrong or d 1 2 3 6 n 13 7 5 3 or B1 for one correct sequence or three pairs from table 5(a) ( ) 2 2 100 10 d d = or k = 10 C1 21 1 5(b)(i) k = d soi C1 n = 2d + 1 1 5(b)(ii) Increasing linear sequence of 11 terms with common difference = 5 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 9 of 11 Hammer Throwing Question Answer Marks Partial Marks Modelling: Hammer Throwing 6(a) 2π × 1.8 C1 11.3 m 1 6(b) V = 11.3N 1 FT their 11.3 7(a) 10 1 m/s or ms–1 C1 7(b)(i) 10x 1 FT their numerical 10 7(b)(ii) 10x + 2 oe 1 FT their expression in x 7(c)(i) ( ) 2 10 2 4.9 H their x x = + − oe 1 FT their (10x + 2) 7(c)(ii) Sketch of 2 10 2 4.9 x x + − for x ⩾ 0 indicating where H = 0 OR 2 4.9 10 2 0 x x − − = oe and sketch of their quadratic intersecting the positive x-axis or sketch of e.g. 2 4.9 y x = and y = 10x + 2 or 2 10 ( 10) 4 4.9 ( 2) 2 4.9 + − − −  oe with square root sign seen for their three-term quadratic equation in terms of x. C2 C1 for sketch of 10x + 2 – 4.9x2 for x ⩾ 0 FT their model for H of form   2 ax bx c + + OR C1 for their H = 0 oe where H is an expression in x. or provided equation given, sketch of their quadratic intersecting the positive x-axis or sketch of e.g. y = 4.9x2 and y = 10x + 2 or 2 10 ( 10) 4 4.9 ( 2) 2 4.9 + − − −  oe with square root sign seen for their three-term quadratic equation in terms of x. 2.22 1 8(a) [Vertical] speed = 0.7V or height = 0.7Vx[+2] leading to 2 4.9 0.7 2 0 x Vx − − = 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 10 of 11 Question Answer Marks Partial Marks 8(b)(i) 2 ( 0.7 ) ( 0.7 ) 4 4.9 ( 2) 2 4.9 −− + − − −  V V oe 1 Correct steps 2 0.7 0.49 4 4.9 2 9.8 + +   V V factor out 0.49 or 0.49 e.g. 2 0.7 0.49 80 9.8 + + V V 0.7 seen as factor in both terms on the numerator and simplifying e.g. 2 0.7 0.7 80 9.8 + + V V leading correctly to 2 80 14 + + V V 2 B1 for a correct step to get the factor 0.49 or 0.49 or 49 OR [Divide by or factor out 0.7] 7x2 – Vx – 2 0.7 [ = 0] oe (1) x = 2 2 4 7 0.7 14 + +  V V (1) = 2 80 14 + + V V (1) 8(b)(ii) 2 80  + V V or 2 2 80 V V  + oe 1 9(a)(i) 14m/s 1 9(a)(ii) 2 20 20 80 14 + + C1 41.86 to 41.91 2 B1 for their 2 20 20 80 14 + + × their14 seen 9(b) 2 80 0.7 14 + + V V V leading to 2 2 80 20 + + V V V 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2025 © Cambridge University Press & Assessment 2025 Page 11 of 11 Question Answer Marks Partial Marks 9(c)(i) Sketch of D = 2 2 80 20 + + V V V and D = 82.98 or sketch of 2 2 80 20 + + V V V – 82.98 oe or 2 2 2 80 1659.6 3319.2 V V = − or better C1 28.5 m/s 1 9(c)(ii) 28.5 ÷ 11.3 C1 FT their 9(c)(i) ÷ their 6(a) 2.5 [turns/s] 1 FT their 9(c)(i) ÷ their 6(a) if answer between 1 and 5