Cambridge IGCSE Mathematics - International 0607 — 2023 Feb/March 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) February/March 2023 1 hour 40 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer both part A (Questions 1 to 3) and part B (Questions 4 to 6). ● 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 [ ]. * 0 5 2 0 0 0 2 2 4 2 * DC (PQ/FC) 318194/2 © UCLES 2023

2 0607/62/F/M/23 © UCLES 2023 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 3) SPLIT NUMBERS (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at numbers that are split into two parts. A start number is split into two parts the units, U and the remaining multiple of ten, T. Examples start number T U 37 30 7 125 120 5 1526 1520 6 1 (a) Complete the table. start number T U T 2 U 2 T U 2- 2 T U - T U T U 2 - - 2 125 120 5 14 400 25 14 375 115 125 34 34 40 2 50 0 2500 50 1 1 22 499 151 0 49 000 000 [5] (b) What do you notice about the start number and the value of T U T U 2 - - 2 for each row of the table in part (a)? … [1]

3 0607/62/F/M/23 © UCLES 2023 [Turn over (c) Copy your answers from part (a) into the shaded columns. Complete the table. start number T U T U 2 2 - T U + T U T U 2 + - 2 125 120 5 14 375 125 115 34 34 40 2 42 38 50 0 50 1 22 499 0 49 000 000 [2] (d) Use your tables from part (a) and part (c) to complete this statement. T U T U 2 + - = 2 … [1] (e) (i) Show that the expansion of ( )( ) T U T U T U is 2 - + - 2 . [1] (ii) Using part (i) write 180 5 2 2 - as the product of two integers. [2]

4 0607/62/F/M/23 © UCLES 2023 2 This question is about T U 2 + 2 . T U + and T U - are not always factors of T U 2+ 2. (a) Use the table to help you investigate the start numbers from 35 to 40. Find whether T U + or T U - or both are factors of T U 2 + 2 for these start numbers. You may not need to use all the columns. start number T U T 2 U 2 35 30 5 900 25 36 30 6 900 36 37 30 7 900 49 38 30 8 900 64 39 30 9 900 81 40 40 0 1600 0 [5] (b) A start number is a multiple of 10. Use algebra to explain why T U + and T U - are always factors of T U + 2 2. [2]

5 0607/62/F/M/23 © UCLES 2023 [Turn over (c) (i) Show that ( ) T U T U TU 2 2 2 2 + = + - . [1] (ii) When T U + is a factor of T U 2 2 + , then T U + must also be a factor of 2TU. Explain why this statement is true. [1] (d) When T U - is a factor of T U 2 2 + , then T U - must also be a factor of 2TU. Use the expansion of ( ) T U 2 - to explain why this statement is true. [3]

6 0607/62/F/M/23 © UCLES 2023 3 This question is about T U - 3 3. (a) Complete the table. start number T U - T U 3 - 3 T U T U 3 - - 3 T TU U 2+ + 2 12 8 124 25 15 525 31 29 931 [3] (b) (i) Use the table to complete the statement below. ( )( T U T TU U 2 - + + = ) 2 … [2] (ii) Use algebra to show that the statement in part (i) is always true. [1]

7 0607/62/F/M/23 © UCLES 2023 [Turn over The modelling task starts on the next page.

8 0607/62/F/M/23 © UCLES 2023 B MODELLING (QUESTIONS 4 to 6) FERRIES (30 marks) You are advised to spend no more than 50 minutes on this part. This task is about the time it takes a ferry to leave an island and travel to the mainland. Jin owns a ferry. 4 The ferry leaves the island when there are 10 passengers on the ferry. In this question, there are already 6 passengers on the ferry. This means the ferry needs 4 more passengers before it leaves. Jin wants to model how long the ferry waits at the island before travelling to the mainland. He assumes the passengers arrive one at a time. (a) Jin uses the one-digit numbers 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 to make a rule. This rule gives the number of minutes between each passenger arriving. One-digit numbers 0 1, 2 3, 4, 5, 6, 7 8, 9 Number of minutes between passengers 1 2 3 4 (i) Jin uses his rule with this table of random numbers to model the time between each passenger arriving. Numbers used for One-digit random numbers Trial 1 9 3 5 0 Trial 2 2 1 7 5 Trial 3 5 8 6 0 Trial 4 2 4 9 6 Trial 5 4 1 7 9 Trial 6 9 3 5 8 Trial 7 5 1 0 8 Trial 8 3 7 5 5 Trial 9 9 5 4 6 Trial 10 3 3 1 7

9 0607/62/F/M/23 © UCLES 2023 [Turn over Each trial models the times between passengers arriving. Use the last two rows of the random number table to complete the table of trials. Jin has completed the first eight trials. Example The first random number for Trial 1 is 9. Jin’s rule gives a time between passengers of 4 minutes. These values 9 and 4 are ringed in each table. Number of minutes between passengers Number of minutes ferry waits to leave Trial 1 4 3 3 1 11 Trial 2 2 2 3 3 10 Trial 3 3 4 3 1 11 Trial 4 2 3 4 3 12 Trial 5 3 2 3 4 12 Trial 6 4 3 3 4 14 Trial 7 3 2 1 4 10 Trial 8 3 3 3 3 12 Trial 9 Trial 10 [2] (ii) Use trials 1 to 10 to work out an estimate of the mean time before the ferry leaves when there are already 6 passengers on the ferry. … [2] (iii) Jin estimates that the total time for the first 6 passengers to arrive is 14.5 minutes. Calculate the mean time for there to be 10 passengers on the ferry. … [2]

10 0607/62/F/M/23 © UCLES 2023 (b) When the ferry leaves the island it travels 4 km to the mainland. In this direction of travel • the speed of the ferry is v km/h • the speed of the water is 3 km/h • the speed of travel = speed of the ferry + speed of the water. (i) When v 13 = , show that the ferry takes 15 minutes to travel from the island to the mainland. [1] (ii) Find a model for the time, t minutes, the ferry takes to travel from the island to the mainland. Give your answer in terms of v, in its simplest form. … [2]

11 0607/62/F/M/23 © UCLES 2023 [Turn over (iii) Sketch a graph of your model for v 0 40 G G . v 40 0 0 t [2] (iv) Find t when the speed of travel is 25 km/h. … [2]

12 0607/62/F/M/23 © UCLES 2023 5 In this question, there are no passengers on the ferry. Jin updates his model in Question 4(a) in the following way: • The ferry is at the island. • The passengers now arrive in groups. • The ferry travels to the mainland as soon as a group arrives making the total number of passengers 10, 11 or 12. (a) (i) Jin uses the two-digit numbers from 00 to 99 to make a second rule. This rule gives the number of passengers in a group. Complete the table. Two-digit numbers 00 to 19 20 to 64 … Number of passengers in group 1 2 3 [1] (ii) He also uses his one-digit rule from Question 4(a) for the time between each group arriving. One-digit numbers 0 1, 2 3, 4, 5, 6, 7 8, 9 Number of minutes between groups 1 2 3 4 Jin uses his rules with these tables of random numbers to model the number of passengers in each group and the time between each group arriving. Numbers used for Two-digit random numbers One-digit random numbers Trial 1 08 89 97 24 71 12 2 3 4 4 9 1 Trial 2 27 92 18 10 42 60 2 4 1 1 7 6 Trial 3 45 99 94 40 68 29 5 0 8 0 6 2 Trial 4 32 77 66 26 45 03 9 2 3 8 5 4 Trial 5 96 09 55 83 21 72 4 2 1 6 7 4 Trial 6 02 22 80 18 83 54 9 0 3 2 5 8 Trial 7 88 07 43 48 06 11 5 3 1 2 0 7 Trial 8 98 66 14 45 15 73 3 1 7 5 5 2 Trial 9 02 21 00 18 63 70 9 0 5 5 4 8 Trial 10 51 99 30 55 05 48 8 8 9 0 3 5

13 0607/62/F/M/23 © UCLES 2023 [Turn over Use the last two rows of each random number table to complete the table of trials. Jin has completed the first eight trials. In Trial 1, when five groups have arrived there is a total of 12 passengers and so the ferry leaves. The sixth group of passengers and the sixth number of minutes are both crossed out. Number of passengers in group Number of minutes between groups Number of minutes ferry waits to leave Trial 1 1 3 3 2 3 1 2 3 3 3 4 2 15 Trial 2 2 3 1 1 2 2 2 3 2 2 3 3 15 Trial 3 2 3 3 2 3 2 3 1 4 1 3 2 9 Trial 4 2 3 3 2 2 1 4 2 3 4 3 3 13 Trial 5 3 1 2 3 2 3 3 2 2 3 3 3 13 Trial 6 1 2 3 1 3 2 4 1 3 2 3 4 13 Trial 7 3 1 2 2 1 1 3 3 2 2 1 3 14 Trial 8 3 3 1 2 1 3 3 2 3 3 3 2 14 Trial 9 Trial 10 [3] (iii) Calculate the difference between the mean number of minutes the ferry waits to leave given by the model in Question 4(a)(iii) and this updated model. … [2] (iv) (a) Give a reason why modelling the arrival of passengers in groups improves the model in Question 4(a). … … [1] (b) Write down one way that Jin could improve his model in Question 5(a). … … [1]

14 0607/62/F/M/23 © UCLES 2023 (b) Jin wants the total time from when the ferry returns to the island to when it next arrives at the mainland to be no more than 20 minutes. He estimates the speed, v, at which the ferry must travel using the model in Question 4(b) and the mean time given by the model in Question 5(a). Find the value of v, the speed of the ferry from the island to the mainland. … [3] 6 Jin changes the route of the ferry so that it makes a stop before it arrives at the mainland. When there are at least 10 passengers, the ferry • starts at the island (I) • travels to another island (A) • waits 16 minutes then • travels to the mainland (M). The diagram shows the ferry journey and the distance, in km, of each part. NOT TO SCALE I A M 1 km 4 km The table shows the speed of the ferry and the speed of the water, in km/h, for each part of the journey. I to A A to M Speed of ferry v v 1 - Speed of water 2 1

15 0607/62/F/M/23 © UCLES 2023 (a) Jin finds a model for the total time, t minutes, the ferry takes from when it returns to the island to when it next arrives at the mainland. He uses the mean time from the model in Question 5(a). His model is ( ) ( ) t v v k v v 2 12 4 2 = + + + where k is an integer. Find the value of k. … [4] (b) For this route, the maximum value of v, the speed of the ferry, is 40 km/h. Jin wants the total time from when the ferry returns to the island to when it next arrives at the mainland to be no more than 35 minutes. Work out if it is possible for the ferry to complete the journey in this time. … [2]

16 0607/62/F/M/23 © UCLES 2023 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

This document consists of 9 printed pages. © UCLES 2023 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) February/March 2023 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 February/March 2023 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 February/March 2023 © UCLES 2023 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 descriptors 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 February/March 2023 © UCLES 2023 Page 3 of 9 Maths-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, 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 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 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 February/March 2023 © UCLES 2023 Page 4 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

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2023 © UCLES 2023 Page 5 of 9 Question Answer Marks Partial Marks INVESTIGATION: SPLIT NUMBERS 1(a) start T U T 2 U 2 T 2 − U 2 T – U 2 2 T U T U − − 125 120 5 14400 25 14375 115 125 34 30 4 900 16 884 26 34 42 40 2 1600 4 1596 38 42 50 50 0 2500 0 2500 50 50 151 150 1 22500 1 22499 149 151 7000 7000 0 49000000 0 49000000 7000 7000 5 B1 for each correct row 1(b) They are the same oe 1 1(c) start T U T 2 − U 2 T + U 2 2 − + T U T U 125 120 5 14375 125 115 34 30 4 884 34 26 42 40 2 1596 42 38 50 50 0 2500 50 50 151 150 1 22499 151 149 7000 7000 0 49000000 7000 7000 2 FT their 884 and their 2500 B1 for four correct cells 1(d) T – U oe 1 1(e)(i) T2 – TU + TU – U2 1 1(e)(ii) (180 – 5)(180 + 5) C1 175  185 isw 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2023 © UCLES 2023 Page 6 of 9 Question Answer Marks Partial Marks 2(a) Table headings including T – U and T2 + U2 C1 Correct answer in the table to a relevant division or division with correct answer below the table C1 Complete their table start T – U T + U T2 + U2 2 2 T U T U + + or T + U factor 2 2 T U T U + − or T – U factor 35 25 35 925 26.4 or No or × 37 or Yes or ✓ 36 24 36 936 26 or Yes or ✓ 39 or Yes or ✓ 37 23 37 949 25.6 or No or × 41.26 or No or × 38 22 38 964 25.368 or No or × 43.8 or No or × 39 21 39 981 25.15 or No or × 46.7 or No or × 40 40 40 1600 40 or Yes or ✓ 40 or Yes or ✓ 3 B2 for one correct column giving an answer to division by T + U or T – U, assuming both if a joint column seen or B1 for one correct column with 4 cells correct 2(b) U = 0 soi as a general result e.g. U will always be 0 1 T 2 + U 2 = T 2 and both T + U and T – U = T. 1 2(c)(i) [(T + U)2 – 2TU =] 2 2 2 2 + + − T TU U TU = T2 + U2 1 2(c)(ii) T + U is a factor of (T + U)2 so therefore T + U is a factor of [–] 2TU 1 2(d) (T – U)2 = T2 + U2 –2TU or T2 + U2 = (T – U)2 + 2TU 2 B1 for 2 2 2 − + T TU U T – U is a factor of (T – U)2 so therefore T – U is a factor of 2TU 1 3(a) One relevant calculation seen or for identifying a correct pair of values for T and U C1 start number T – U T 3 − U 3 3 3 T U T U − − T 2 + TU + U 2 12 8 992 124 124 25 15 7875 525 525 31 29 26 999 931 931 2 B1 for each column

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2023 © UCLES 2023 Page 7 of 9 Question Answer Marks Partial Marks 3(b)(i) 3 3 − − T U T U = T 2 + TU + U 2 or 124  8 = 992 or 525 × 15 = 7875 or 931 × 29 = 26999 C1 T 3 − U 3 1 3(b)(ii) 3 2 2 2 2 3 3 3 + + − − − = − T T U TU UT TU U T U 1 MODELLING: FERRIES 4(a)(i) Trial 9 4 3 3 3 13 Trial 10 3 3 2 3 11 2 B1 for each correct row 4(a)(ii) 11.6 1 FT their 13 and their 11 minutes or min C1 4(a)(iii) 14.5 + 11.6 C1 FT their 11.6 26.1 min 1 FT their 11.6 4(b)(i) 4 60 15 16  = or 4 1 16 4 = hour = 15 min 1 4(b)(ii) 240 3 = + t v 2 B1 for distance speed used e.g 4 3 + v seen 4(b)(iii) Correct sketch 40 t v 0 1 Correct shape not touching or crossing v axis t-intercept labelled as 80 C1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2023 © UCLES 2023 Page 8 of 9 Question Answer Marks Partial Marks 4(b)(iv) [v =] 22 seen in working or on sketch or 240 25 or 4 25 seen C1 9.6 min 1 If 0 scored for 4(b)(iv) SC1 for 8.57 5(a)(i) 65 to 99 1 5(a)(ii) Number of passengers in group Number of minutes between groups 1 2 1 1 2 3 4 1 3 3 3 4 18 2 3 2 2 1 2 4 4 4 1 3 3 16 3 B1 for 2 correct sections B1 for 18 or 16 or 19 5(a)(iii) 14 1 FT their 18 and their 16 12.1 1 FT their 26.1 – their 14 5(a)(iv) (a) Groups are more likely oe 1 5(a)(iv) (b) e.g. Run more trials Increase the number of passenger groups to more than 3 Increase the number of minutes between groups to more than 4 Use two-digit numbers for number of minutes between groups Use three-digit numbers for number of passengers in group 1 5(b) 20 240 14 3 = + + v oe or horizontal line at t = 6 or at t = 20 drawn on appropriate sketch C1 FT 20 = 240 14 3 + + their their v Correct rearrangement of their equation to isolate v or an equation with v coefficient = 1 or horizontal line at t = 6 or at t = 20 labelled correctly C1 37 nfww 1 FT their equation dep on first C1 SC1 for 9 following 20 240 3 = + v

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2023 © UCLES 2023 Page 9 of 9 Question Answer Marks Partial Marks 6(a) [t =] 240 60 16 14 2 + + + + their v v oe C2 C1 for   4 1 60 2 1 1   +    + −+   v v or three of the four terms correct t = 240 60( 2) 30 ( 2) ( 2) + + + + + v v v v v v or better C1 30 nfww 1 6(b) 2 30(40 12(40) 4) 40(42) + + = t oe or ( )( ) ( ) 2 30 12 4 35 2 + + = + their v v v v oe C1 FT their30 No and t = 37[.21… minutes] or No and v = 58[.41… km/h] 1 FT their30