Cambridge IGCSE Mathematics - International 0607 — 2024 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 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 7) and part B (Questions 8 to 10). ● 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 [ ]. * 9 1 2 4 5 2 5 7 2 0 * DC (PQ/SG) 332231/1 © UCLES 2024

2 0607/62/F/M/24 © UCLES 2024 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 7) MAGIC SQUARES (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at ways to make a magic square. A magic square is a grid with a different number in each square. The numbers in each row, each column or each diagonal all add up to the same total. This is the line total. Example This is a 3 by 3 magic square using the integers 1 to 9. The line total is 15. 1 2 Column 3 1 2 Row 3 Left diagonal Right diagonal 1 8 6 5 3 7 9 4 2 Line totals Row 1 8 1 6 15 + + = Column 1 8 15 3 4 + + = Left diagonal 6 15 4 5 + + = Row 2 15 3 5 7 + + = Column 2 15 1 5 9 + + = Right diagonal 8 15 5 2 + + = Row 3 15 4 9 2 + + = Column 3 15 6 7 2 + + =

3 0607/62/F/M/24 © UCLES 2024 [Turn over 1 A 3 by 3 magic square uses the integers 1 to 9. Complete the following statements to show that the line total is 15. The total of the integers 1 to 9 = … The total of the integers in all three rows = … The line total = … ' … = 15 [2] 2 (a) This is a different 3 by 3 magic square using the integers 1 to 9. Complete this magic square. 4 5 1 [2] (b) There are 8 different 3 by 3 magic squares using the integers 1 to 9. In all these magic squares the integer in the middle square is the same. (i) What is the connection between the integer in the middle square and the integers 1 to 9? … … [1] (ii) Explain how to use the integer in the middle square to find the line total. … [1]

4 0607/62/F/M/24 © UCLES 2024 3 A and B are two magic squares using the integers 1 to 9. More magic squares can be made using reflection. B is made by reflecting the position of each integer using a vertical line of reflection. The line of reflection goes through the middle column of A. The numbers move from the square in A to the reflected square in B. 1 A B 8 6 5 3 7 9 4 2 1 8 6 5 3 7 9 4 2 1 6 8 5 7 3 9 2 4 Line of reflection (a) Draw a horizontal line of reflection through the middle row of magic square A. Complete the new magic square C using the horizontal line of reflection. 7 3 5 C [1]

5 0607/62/F/M/24 © UCLES 2024 [Turn over (b) Two different magic squares can be made by reflection in the left and right diagonal lines drawn on A. Complete these magic squares. Reflection in left diagonal Reflection in right diagonal 6 4 5 5 [2]

6 0607/62/F/M/24 © UCLES 2024 4 This is a method to make an n by n magic square where n is odd. Step 1 Place the smallest integer in the middle square of the top row. Step 2 Move up and right in the direction of the left diagonal to place the next integer. (i) If you go off the magic square, place the next integer at the opposite end of the row or column you are on. (ii) If the square that is up and right has an integer in it, go down one square to place the next integer. (iii) If you are in the top right corner, go down one square to place the next integer. Step 3 Repeat step 2 until the magic square is complete. Example using integers 1 to 9 • Put 1 in the middle square of the top row. • Move up and right. This is off the square, so go to the opposite end of this column and place 2. 1 2 2 • Move up and right from 2. This is off the square, so go to the opposite end of this row and place 3. • Move up and right from 3. This square has 1 in it, so go down one square from 3 and place 4. 1 2 4 3 • Move up and right and place 5 and 6. • 6 is in the top right corner so go down one square and place 7. 1 2 4 3 5 7 6 • Move up and right from 7. This is off the square, so go to the opposite end of this row and place 8. • Move up and right from 8. This is off the square, so go to the opposite end of this column and place 9. 1 8 2 9 4 3 5 7 6

7 0607/62/F/M/24 © UCLES 2024 [Turn over (a) Use the method to make a 3 by 3 magic square with the integers 15 to 23. Step 1 has been done for you. 15 [2] (b) (i) Use the method to complete the 5 by 5 magic square with the integers 1 to 25. The first six numbers have been placed for you. 1 5 6 4 2 3 [2] (ii) Find the line total for this magic square. … [1] (iii) What is the connection between the line total and the integer in the middle square? … [1] (c) A 7 by 7 magic square uses the integers 1 to 49. Find the integer in the middle square and the line total. Integer in the middle square = … Line total = … [2]

8 0607/62/F/M/24 © UCLES 2024 5 This is an expression for the line total in an n by n magic square using the integers 1 to n2. n n 2 1 2 + ` j Find the line total for a magic square using the integers 1 to 81. … [2] 6 Different 3 by 3 magic squares are made using the first nine terms of a sequence. The first term of the sequence is 2. The nth term of the sequence is n 3 1 - . Find six of these magic squares. Use the method of Question 4 and reflection to help you. [5]

9 0607/62/F/M/24 © UCLES 2024 [Turn over 7 When n is even there is no middle square so we use a different method to make a magic square. • Draw the two diagonals on the 4 by 4 square. • Put the smallest integer in the top left square. • Go from left to right, but only write the integers in the squares where there is part of a diagonal. 3 4 7 8 11 12 15 16 6 10 9 5 1 2 14 13 • Starting with the bottom row write the remaining integers, in order, from right to left in the empty squares. 14 4 7 9 11 5 2 16 6 10 8 12 1 15 3 13 (a) Use this method to complete the 4 by 4 magic square with the consecutive integers k to k 15 + . There is an extra grid for working. [2]

10 0607/62/F/M/24 © UCLES 2024 (b) (i) Find the line total in terms of k. Give your answer in its simplest form. … [2] (ii) The line total is 254. Find the value of the largest integer in the third row. … [2]

11 0607/62/ F /M /24 © UCLES 2024 [Turn over B MODELLING (QUESTIONS 8 to 10) DELIVERIES BY SCOOTER (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at average speeds of deliveries made by scooter. A company delivers orders from its warehouses to shops and houses. All deliveries of small parcels are made by scooter. 8 (a) Meera rides a scooter that travels 12 km in 15 minutes on quiet roads. Show that her average speed on quiet roads is 48 km/h. [2] (b) When the roads are busy the time for the 12 km journey is 30 minutes. Calculate her average speed on busy roads. … [2] (c) Meera makes a delivery. She travels for 15 minutes on quiet roads and then for 30 minutes on busy roads. Calculate her average speed for the whole journey. … [2]

12 0607/62/F/M/24 © UCLES 2024 (d) A warehouse is outside the town. In her journey from the warehouse Meera always travels the first 4 km on quiet roads. She then travels x km on busy roads to make her first delivery. (i) Show that a model for S, her average speed in km/h, for this whole journey is S x x 12 1 24 4 = + + . [2] (ii) Show that the model in part (i) simplifies to S x x 2 96 24 = + + . [2] (e) To make her first delivery Meera leaves the warehouse and travels a total of 10 km on quiet roads and busy roads. Use the model in part (d)(ii) to find Meera’s average speed. … [3]

13 0607/62/F/M/24 © UCLES 2024 [Turn over (f) Sketch the model for S in part (d)(ii) on the axes, for x 0 20 G G . 0 20 0 x Distance on busy roads (km) Average speed for whole journey (km/h) S [2] (g) Find the total distance of her delivery journey from the warehouse when her average speed is 27 km/h. … [2]

14 0607/62/F/M/24 © UCLES 2024 9 A second warehouse is in the town. Meera travels at the same speeds on quiet roads as in Question 8(a) and the same speed on busy roads as in Question 8(b). In her journey from this warehouse Meera always travels the first 3 km on busy roads. She then travels d km on quiet roads to make her first delivery. (a) Find a model for her average speed, Y km/h, when making a delivery from this warehouse. Do not simplify your model. … [2] (b) Show that your model in part (a) simplifies to Y d d 6 144 48 = + + . [1] (c) Find the average speed when S Y = and . x d = … [2]

15 0607/62/F/M/24 © UCLES 2024 10 The model for one delivery from the first warehouse, S x x 2 96 24 = + + , is the same as ( ) S x x 2 24 4 = + + . This model can be changed when there are more deliveries. Each delivery is a distance of x km on busy roads from the previous delivery. (a) Rewrite this model for a total of 3 deliveries. … [1] (b) The average time that Meera stops to complete each delivery is 5 minutes. Meera rewrites the model in Question 8(d)(i) to include the times stopped to make all 3 deliveries. Show that her model is ( ) S x x 8 3 24 4 3 = + + . [3] (c) Meera needs to shorten the average time that she stops so that her average speed for the whole journey is 24 km/h. Find the total length of time she now stops to make the 3 deliveries. … [4]

16 0607/62/F/M/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

This document consists of 9 printed pages. © Cambridge University Press & Assessment 2024 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) February/March 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 February/March 2024 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 2024 © Cambridge University Press & Assessment 2024 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 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 February/March 2024 © Cambridge University Press & Assessment 2024 Page 3 of 9 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. 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 2024 © Cambridge University Press & Assessment 2024 Page 4 of 9 Question Answer Marks Partial Marks Investigation MAGIC SQUARES 1 The total of the integers from 1 to 9 = 45 The total of the integers in all three rows = 45 The line total 45 ÷ 3 = 15 2 B1 for 45 seen correctly placed once 2(a) 2 9 4 7 5 3 6 1 8 2 B1 for 9 and 6 correctly placed 2(b)(i) Integer in middle square = middle integer of 1 to 9 oe 1 2(b)(ii) [line total =] integer in middle square × number of rows/columns oe 1 3(a) 4 9 2 3 5 7 8 1 6 1 3(b) Left diagonal Right diagonal 8 3 4 1 5 9 6 7 2 2 7 6 9 5 1 4 3 8 2 B1 for each correct magic square 4(a) 22 15 20 17 19 21 18 23 16 2 B1 for 16, 17 and 18 correctly placed or for 19 in the middle

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 5 of 9 Question Answer Marks Partial Marks 4(b)(i) 17 24 1 8 15 23 5 7 14 16 4 6 13 20 22 10 12 19 21 3 11 18 25 2 9 2 B1 for 13 in the middle 4(b)(ii) 65 1 4(b)(iii) Line total = 5 × the integer in the middle square oe 1 4(c) [Integer in the middle square =] 25 [Line total =] 175 2 B1 for each 5 9(81 1) 2 + C1 369 1 6 [2,] 5, 8, 11, 14, 17, 20, 23, 26 C1 At least two lines of reflection drawn C1 Six from 23 2 17 17 2 23 11 26 5 8 14 20 20 14 8 8 14 20 11 26 5 5 26 11 23 2 17 5 20 17 23 8 11 11 8 23 26 14 2 2 14 26 26 14 2 11 8 23 17 20 5 5 20 17 5 26 11 17 20 5 20 14 8 2 14 26 17 2 23 23 8 11 3 FT their sequence B2 for four correct or B1 for two correct If 0 excluding communication scored SC2 for 4 squares correctly reflected from one of their squares or SC1 for 2 squares correctly reflected from one of their squares

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 6 of 9 Question Answer Marks Partial Marks 7(a) k k + 14 k + 13 k + 3 k + 11 k + 5 k + 6 k + 8 k + 7 k + 9 k + 10 k + 4 k + 12 k + 2 k + 1 k + 15 2 B1 for any 1 row or 1 column correct 7(b)(i) Four expressions from one row or column or diagonal added or all 16 cells added and divide by 4 C1 FT their expressions in k in 7(a) 4k + 30 isw 1 7(b)(ii) 254 = 4k + 30 oe followed by one correct line of working e.g. 224 = 4k or 254 30 4 − C1 FT their (b)(i) in linear k (= 4k + 30) 66 1 FT their 56 substituted into their largest expression in row 3

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 7 of 9 Question Answer Marks Partial Marks Modelling DELIVERIES BY SCOOTER 8(a) 12 ÷ 15 60 oe or 12 60 15  [=48] 2 B1 for 15 60 seen 8(b) 24 1 km/h C1 8(c) 12 12 45 60 + oe or 24 60 45  C1 32 km/h 1 8(d)(i) [Total] distance = 4 + x [Time on] quiet [roads] = 4 48 or 5 60 [Time on] busy [roads] = 24 x 2 B1 for one statement 8(d)(ii) 4 1 12 24 x x + + with correct use of × 24 seen oe with correct working leading to 96 24 2 x S x + = + 2 B1 for × 24 oe or 24 or 48 or 288 oe as correct common denominator 8(e) 10 – 4 [= 6 ] or x = 6 C1 Substitution of 6 into 96 24 2 x x + = + or ( ) ( ) 24 10 2 10 4 + − C1 FT their 6 30 km/h 1 If 0 or only second C1 scored then also SC1 for answer of 28

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 8 of 9 Question Answer Marks Partial Marks 8(f) Correct sketch 1 Correct shape Must reach S axis Should go to approx. halfway between their S axis intersection and 0 48 marked at S axis intersection C1 8(g) Line on sketch at S = 27 C1 18 km 1 9(a) 3 3 24 48 d Y d + = + isw oe 2 FT their 24 in 8(b) B1 for either 3 24 or 48 d as part of denominator 9(b) Y = 3 or 6 48 48 d Y d + = + 3 6 48 d d + + oe or correct use of × 48 oe leading to Y = 144 48 6 d d + + . 1 9(c) Correct sketch or 96 24 2 x x + + = 144 48 6 x x + + and 576 + 144x +96x +24x2 = 288 +96x +144x +48x2 or better C1 32.8 km/h 1 10(a) ( ) 24 4 3 2 3 x S x + = + oe 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 9 of 9 Question Answer Marks Partial Marks 10(b) 4 3 1 3 15 12 24 60 x x + + + oe 2 B1 for 15 seen or 15 60   =     6 24 or 1 4 or 3( 1 12 ) or 1 4 Correct clearance of fractions in denominator leading to S = ( ) 24 4 3 8 3 x x + + 1 10(c) 4 3 1 3 12 24 60 x x t + + + = 24 or better or 4 3 24 1 3 12 24 x x t + = + + or better or Total time = 4 3 24 x + C1 t can be any variable Correct removal of fractions from denominator or subtract 3 24 x and subtract 1 12 C1 5 [min] or 2 24 [h] oe 1 Correct units dependent on use of variable, e.g. t C1