Cambridge IGCSE Mathematics - International 0607 — 2024 Oct/Nov Paper 6 · Variant 3

This document has 20 pages. Any blank pages are indicated. [Turn over * 8 1 1 8 0 0 2 9 2 5 * Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 Investigation and Modelling (Extended) October/November 2024 1 hour 40 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer both part A (Questions 1 to 6) and part B (Questions 7 to 9). ● 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 60. ● The number of marks for each question or part question is shown in brackets [ ]. DC (DE/SW) 337392/3 © UCLES 2024 ,










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2 0607/63/O/N/24 © UCLES 2024 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 6) DIAGONAL SQUARES (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at the number of diagonal squares in a grid where the diagonals of each square have been drawn. This grid is h by w, where h is the number of rows in the grid and w is the number of columns in the grid. w h Example This grid is 3 by 5. It has 3 rows and 5 columns, so h 3 = and w 5 = . Some of the diagonal squares are shaded. Throughout the investigation the number of diagonal squares in an h by w grid is called D(h, w). 1 This question is about grids with h 1 = . A 1 by 2 grid has one diagonal square, so D(1, 2) = 1. A 1 by 3 grid has two diagonal squares, so D(1, 3) = 2. A 1 by 4 grid has three diagonal squares, so D(1, 4) = 3. * 0000800000002 * ,










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3 0607/63/O/N/24 © UCLES 2024 [Turn over (a) Write down D(1, 7). … [1] (b) Find an expression for D(1, w). … [1] (c) D(1, w) = 20. Write down the value of w. … [1] (d) Explain why D(1, n) = D(n, 1). … [1] * 0000800000003 * ,










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4 0607/63/O/N/24 © UCLES 2024 2 This question is about grids with h 2 = . A 2 by 2 grid has 2 rows each with 1 diagonal square and 1 row of 2 diagonal squares. The table shows this. 2 by 2 grid Number of rows of diagonal squares Number of diagonal squares in each row Total 2 1 2 1 2 # = 1 2 2 1 2 # = D(2, 2) = 2 2 4 + = A 2 by 4 grid has 2 rows of 3 diagonal squares and 1 row of 4 diagonal squares. The table shows this. 2 by 4 grid Number of rows of diagonal squares Number of diagonal squares in each row Total 2 3 2 3 6 # = 1 4 1 4 4 # = D(2, 4) = 6 4 10 + = * 0000800000004 * ,










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5 0607/63/O/N/24 © UCLES 2024 [Turn over (a) Complete the table. You may use the grids to help you. 2 by 8 grid Number of rows of diagonal squares Number of diagonal squares in each row Total 2 1 D(2, 8) = + = 22 [3] (b) A 2 by w grid has h 2 = and width w. Complete the table with expressions in terms of w. 2 by w grid Number of rows of diagonal squares Number of diagonal squares in each row Total 2 1 D(2, w) = [3] * 0000800000005 * ,










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6 0607/63/O/N/24 © UCLES 2024 3 This question is about grids with h 3 = . A 3 by 5 grid has 3 rows each with 4 diagonal squares and 2 rows each with 5 diagonal squares. The table shows this. 3 by 5 grid Number of rows of diagonal squares Number of diagonal squares in each row Total 3 4 3 4 12 # = 2 5 2 5 10 # = D(3, 5) = 12 10 22 + = (a) Work out D(3, 8). … [2] * 0000800000006 * ,










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7 0607/63/O/N/24 © UCLES 2024 [Turn over (b) A 3 by w grid has h 3 = and width w. Complete the table with expressions in terms of w. 3 by w grid Number of rows of diagonal squares Number of diagonal squares in each row Total 3 2 D(3, w) = [3] (c) In a 3 by w grid, D(3, w) = 97. Find the value of w. … [2] * 0000800000007 * ,










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8 0607/63/O/N/24 © UCLES 2024 4 In an h by w grid there are: • h rows with ( ) w 1 - diagonal squares • ( ) h 1 - rows with w diagonal squares. Show that D(h, w) = ( ) hw h w 2 - + . [2] * 0000800000008 * ,










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9 0607/63/O/N/24 © UCLES 2024 [Turn over 5 This question is about a square grid with n rows and n columns. This is an n by n grid. (a) D(h, w) = ( ) hw h w 2 - + Show that the number of diagonal squares in an n by n grid, D(n, n), is always an even number. [2] (b) The width of square grid A is one more than the width of square grid B. The width of square grid B is n. The difference between the number of diagonal squares in the two square grids is 36. Find the value of n. … [4] * 0000800000009 * ,










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10 0607/63/O/N/24 © UCLES 2024 6 A grid is a rectangle. It has 31 diagonal squares. D(h, w) = ( ) hw h w 2 - + Find the sizes of the possible rectangular grids when h w 1 . … [5] * 0000800000010 * ,










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11 0607/63/O/N/24 © UCLES 2024 [Turn over B MODELLING (QUESTIONS 7 TO 9) SAFE OVERTAKING (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at the distance travelled by a car when overtaking a truck. 7 The table shows the distance of a car from a fixed point on the road at time t seconds. Time (t seconds) 0 1 2 3 4 5 Distance (d metres) 10 12 14 16 18 20 The graph shows this information. 0 0 5 10 Distance (metres) Time (seconds) 15 20 1 2 3 4 5 d t (a) Find the speed of the car. … [2] (b) Find a model for d in terms of t. … [2] (c) Find the distance of the car from the fixed point when t 7 = seconds. … [1] * 0000800000011 * ,









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12 0607/63/O/N/24 © UCLES 2024 8 A car and a truck travel in the same direction along a road. The sketch graph shows their distances from a fixed point on the road. Distance (metres) Time (seconds) 0 45 P Truck Car 0 d t (a) Write down the number of metres the truck is in front of the car when t 0 = . … [1] (b) A model for the distance the car travels, in terms of t, is d t 25 = . The speed of the truck is 20 m/s. Find a model for the distance the truck travels in terms of t. … [1] * 0000800000012 * ,









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13 0607/63/O/N/24 © UCLES 2024 [Turn over (c) The car overtakes the truck at point P. Use algebra to find the coordinates of point P. ( … , … ) [3] (d) When t 5 = the car changes its speed to 30 m/s. The two lines now meet at a different point. (i) On the axes, sketch the new graph for the car. [1] (ii) Find the number of seconds the car has travelled from the fixed point when it reaches the truck. … [4] * 0000800000013 * ,









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14 0607/63/O/N/24 © UCLES 2024 9 This question is about what happens when a car overtakes a truck. The diagrams are not to scale. In these diagrams, the direction of travel is from left to right. The car has length 5 metres. The truck has length 15 metres. The car moves into the outside lane to overtake the truck when there is a gap of 45 metres between the front of the car and the back of the truck. Outside lane Inside lane 5 m 15 m 45 m The car moves back into the inside lane when there is a gap of 50 metres between the back of the car and the front of the truck. Outside lane Inside lane 5 m 15 m 50 m (a) When the car starts overtaking the distance between the front of the car and the front of the truck is 60 m. Show that when the car finishes overtaking the distance between the front of the car and the front of the truck is 55 m. [1] * 0000800000014 * ,









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15 0607/63/O/N/24 © UCLES 2024 [Turn over (b) The car starts overtaking the truck at time t 8 = seconds. The car finishes overtaking the truck at time t k = seconds. The sketch graph shows their distances from a fixed point on the road. Distance (metres) Time (seconds) 0 100 Truck (40,900) Car 0 8 k 55 m 60 m d t The distance travelled, d metres, and the time taken, t seconds, are measured from a fixed point on the road. A model for the distance travelled by the car, in terms of t, is d t 25 = . Find the speed of the truck and write down a model for the distance travelled by the truck. Speed … Model … [3] * 0000800000015 * ,









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16 0607/63/O/N/24 © UCLES 2024 (c) When t 8 = , the truck is 60 metres in front of the car. Find the time, k seconds, when the car is 55 metres in front of the truck. … [2] (d) Work out the total distance that the car travels when overtaking the truck. … [2] (e) When the car overtakes the truck it travels further than the truck. (i) Show that the difference between the distances travelled by the car and the truck is 115 metres. [1] * 0000800000016 * ,









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17 0607/63/O/N/24 © UCLES 2024 (ii) The speed of the car is now x m/s and the speed of the truck is 20 m/s. Write down an expression for the difference in speed between the car and the truck. … [1] (iii) Show that a model for the total distance, D metres, the car travels when overtaking the truck is D x x 20 115 = - . [2] (iv) The car travels 690 metres when overtaking the truck. Find the speed of the car when overtaking the truck. … [3] * 0000800000017 * ,









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18 0607/63/O/N/24 © UCLES 2024 BLANK PAGE * 0000800000018 * ,









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20 0607/63/O/N/24 © UCLES 2024 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 * 0000800000020 * ,










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This document consists of 8 printed pages. © Cambridge University Press & Assessment 2024 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) October/November 2024 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 2024 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 2024 © Cambridge University Press & Assessment 2024 Page 2 of 8 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 2024 © Cambridge University Press & Assessment 2024 Page 3 of 8 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 2024 © Cambridge University Press & Assessment 2024 Page 4 of 8 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 Investigation DIAGONAL SQUARES 1(a) 6 1 1(b) w – 1 1 1(c) 21 1 1(d) [90] rotation oe 1 2(a) 2 7 14 1 8 8 D(2, 8) = 14 + 8 = 22 2 B1 for three correct cells. 2 × 7 or 1 × 8 or Indication of correct 14 squares or 8 squares on grids C1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2024 © Cambridge University Press & Assessment 2024 Page 5 of 8 Question Answer Marks Partial Marks 2(b) 2 w – 1 2w – 2 oe 1 w w D(2, w) = 3w – 2 oe isw 3 B1 for first row. B1 for second row. B1FT for third row. 3(a) 3 × 7 and 2 × 8 or 21 and 16 or Indication of correct 7 squares and correct 8 squares on grids C1 FT 3 × their 7 + 2 × their 8 in 2(a) 37 1 3(b) 3 w – 1 3w – 3 oe 2 w 2w D(2, w) = 5w – 3 oe isw 3 B1 for first row. B1 for second row. B1FT for third row. 3(c) 5w – 3 = 97 C1 FT their (5w – 3) 20 1 4 ( ) ( ) 1 and 1  − −  h w h w oe 1 leading to − + − hw h hw w ( ) 2 − + hw h w 1 5(a) 2 2 2 − n n 1 has a factor of 2 so must be/always even or is a multiple of 2 so must be/always even oe 1 Dep on first mark 5(b) n + 1 seen C1 ( ) ( ) 2 2 2 2 and 2 1 2 1 − + − + n n n n oe seen C1 FT correct substitution of n + 1 in their 5(a) if quadratic in n Correct simplification to   4 36 = n C1 9 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2024 © Cambridge University Press & Assessment 2024 Page 6 of 8 Question Answer Marks Partial Marks 6 Two correct substitutions of h and w with h < w or Two correct substitutions of h and w with h < w in h w – 1 h(w – 1) h – 1 w w(h – 1) with totals or Two correct substitutions of h or w and solution of equation = 31 C2 C1 for one correct substitution of h and w with h < w or one correct substitution of h and w with h < w in h w – 1 h(w – 1) with total h – 1 w w(h – 1) one correct substitution of h or w and solution of equation = 31 If C0 scored SC1 for one correct diagram with correct shading and correct total 1 by 32 oe 2 by 11 oe 4 by 5 oe 3 B1 for each

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2024 © Cambridge University Press & Assessment 2024 Page 7 of 8 Question Answer Marks Partial Marks Modelling 7(a) 2 1 m/s C1 7(b) 2t + 10 1 FT their 2 = d C1 7(c) 24 m 1 8(a) 45 1 8(b)   20 45 = + d t cao 1 8(c) 25 = t 20 45 + t oe C1 FT their ( ) 20 45 + t ( ) 9, 225 cao 2 B1 for each coordinate 8(d)(i) A steeper straight line starting from the car’s line at approx. t = 5 1 8(d)(ii) 145 – 125 or ( ) 125 30 5 + − t oe or 45 + 20(5 + x) = 25 × 5 + 30x oe C2 C1 for 125 or 145 C1 for either expression 20 10 or ( ) 20 45 125 30 5 + = + − t t oe or [5 +] 2 C1 FT 20 30 20 − their FT their ( ) 20 45 + t OR Three complete correct trials of 45 + 20(5 + x) = 25 × 5 + 30x C3 C2 for 2 complete correct trials or C1 for 1 complete correct trial 7 1 9(a) 5 50 + [= 55] 1 9(b)   900 1 00 40 0 − − or 260 1 00 8 − C1 d = 20t + 100 2 B1 for [speed =] 20 m/s

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2024 © Cambridge University Press & Assessment 2024 Page 8 of 8 Question Answer Marks Partial Marks 9(c) 20 1 00 + k = 25 – 55 k oe or 60 55 25 20 + − or List of pairs of distances including 720 with 775 C1 FT their 20 1 00 + t 31 1 9(d) 25 × 8 or 25 × 31 or 25 × (31 – 8) C1 FT their 9(c) for 31 575 m 1 FT their 9(c) if answer positive 9(e)(i) 60 + 55 or 5 × (31 – 8) or 575 – (20 × (31 – 8)) or 575 – ((20 × 31+ 100) – 260) 1 9(e)(ii) x – 20 1 9(e)(iii) s[peed] × t[ime] = x × 115 20 − x leading to 115 20 − x x 2 B1 for t[ime] = 115 20 − x 9(e)(iv) 115 20 − x x = 690 or 20 : 575 x : 690 oe or sketch of positive branch of D = 115 20 − x x C1 FT their 575 in 9(d) 690(x – 20) = 115x or better or 20 690 575  oe or horizontal line at 690 C1 FT their 575 in 9(d) 24 m/s 1