Cambridge IGCSE Mathematics - International 0607 — 2024 Feb/March Paper 5 · Variant 2

This document has 8 pages. [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 Paper 5 Investigation (Core) February/March 2024 1 hour 10 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 36. ● The number of marks for each question or part question is shown in brackets [ ]. * 2 7 4 0 0 2 7 1 0 1 * DC (PQ/JG) 332261/3 © UCLES 2024

2 0607/52/F/M/24 © UCLES 2024 Answer all the questions. INVESTIGATION MAGIC SQUARES 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 3 1 2 3 Right diagonal Left diagonal Row Column 8 1 6 3 5 7 4 9 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 5 15 8 2 + + = Row 3 15 4 9 2 + + = Column 3 15 6 7 2 + + = 1 This is part of a different 3 by 3 magic square using the integers 1 to 9. 2 9 4 7 3 1 (a) Work out the line total for this magic square. … [2] (b) Complete this magic square. [3]

3 0607/52/F/M/24 © UCLES 2024 [Turn over 2 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. 8 1 6 3 4 2 5 7 9 8 1 6 3 4 2 5 7 9 6 1 8 7 2 4 5 3 9 Line of reflection A B (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. 5 3 7 C [1] (b) Two different magic squares can be made by reflection in the left and right diagonal lines drawn on A. Complete these magic squares. 5 4 6 5 Reflection in left diagonal Reflection in right diagonal [2]

4 0607/52/F/M/24 © UCLES 2024 3 (a) (i) Calculate the total of the integers 1 to 9. … [1] (ii) A 3 by 3 magic square uses the integers 1 to 9. Complete the following statements to find the line total for this magic square. The total of the integers in all three rows = … The line total = … ' … = … [2] (b) In all of the magic squares using the integers 1 to 9 the integer in the middle square is the same. (i) Write down the integers 1 to 9. … [1] (ii) What is the connection between the middle integer of the integers 1 to 9 and the magic square? … … [1] (iii) Explain how to use the middle integer of the integers 1 to 9 to find the line total of a 3 by 3 magic square. … [1]

5 0607/52/F/M/24 © UCLES 2024 [Turn over 4 A 3 by 3 magic square is made using the even integers from 2 to 18. (a) (i) Find the middle integer of these integers. … [2] (ii) Use your answer to Question 3(b)(iii) to find the line total for this magic square. … [1] (iii) Use your answers to part (i) and part (ii) to help you complete the magic square below. 12 16 [2] (b) Use your answer to part (a)(iii) and a reflection to make another 3 by 3 magic square. [1]

6 0607/52/F/M/24 © UCLES 2024 5 This is a method to make a 3 by 3 magic square. Example This uses the integers 1 to 9. Step 1 Add one square to each edge to make a cross shape. Step 2 Draw three diagonals on the grid. Each one goes through three squares. Step 3 Write the smallest integer in the bottom square. Write the other integers along the diagonals in order, as shown. Step 4 Move the integers from the outside squares to the empty squares inside and opposite. Completed magic square 9 9 8 8 6 2 4 4 2 3 7 5 1 1 6 5 3 7 8 4 2 1 6 5 9 3 7

7 0607/52/F/M/24 © UCLES 2024 [Turn over (a) (i) Use the method to make a 3 by 3 magic square using the integers 15 to 23. [4] (ii) Work out the line total. … [1] (b) (i) You can use a similar method to make a 5 by 5 magic square using the integers 1 to 25. Two of the diagonals through 5 squares have been drawn. The first six integers have been placed on the correct diagonals. Complete the 5 by 5 magic square. The integers 1, 3 and 19 have been put in their final positions to help you. Do not put integers in the shaded squares. 19 1 5 4 3 2 1 6 [5] (ii) Complete this statement. The line total is … = … # middle integer. [2] Question 6 is printed on the next page.

8 0607/52/F/M/24 © UCLES 2024 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 four of these magic squares. Use the method of Question 5 and reflection to help you. [4] 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.

This document consists of 7 printed pages. © Cambridge University Press & Assessment 2024 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 Paper 5 (Core) February/March 2024 MARK SCHEME Maximum Mark: 36 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.

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 2 of 7 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.

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 3 of 7 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

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 4 of 7 Question Answer Marks Partial Marks 1(a) 15 1 2 + 9 + 4 C1 1(b) 2 9 4 7 5 3 6 1 8 3 B1 for each of 5, 6 and 8 correctly placed 2(a) 4 9 2 3 5 7 8 1 6 1 2(b) Left diagonal Right diagonal 2 7 6 9 5 1 4 3 8 8 3 4 1 5 9 6 7 2 2 B1 for each correct magic square 3(a)(i) 45 1 3(a)(ii) The total of the integers in all three rows = 45 The line total = 45 ÷ 3 = 15 2 B1 for second line correct or for their 45 ÷ 3 3(b)(i) 1, 2, 3, 4, 5, 6, 7, 8, 9 1 3(b)(ii) Middle integer of 1 to 9 = middle integer of magic square 1 3(b)(iii) [line total =] middle integer × 3 oe 1 4(a)(i) 2, 4, 6, 8, 10, 12, 14, 16, 18 or 2 18 2 + C1 10 1 4(a)(ii) 30 1 FT their 10 4(a)(iii) 16 2 12 6 10 14 8 18 4 2 B1 for their 30 – 28

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 5 of 7 Question Answer Marks Partial Marks 4(b) One of these 8 18 4 12 2 16 6 10 14 14 10 6 16 2 12 4 18 8 16 6 8 4 14 12 2 10 18 18 10 2 12 14 4 8 6 16 1 FT their magic square in 4(a)(iii) 5(a)(i) Extra boxes drawn with correct numbers or three diagonals drawn i.e. or 23 21 17 15 C1 22 15 20 17 19 21 18 23 16 3 B2 for any four numbers correctly placed or B1 for 19 in the middle 5(a)(ii) 57 1

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 6 of 7 Question Answer Marks Partial Marks 5(b)(i) Three of 10, 16, 20, 21, 22, 24 or 25 in correct outside squares 25 24 20 22 10 21 5 16 4 6 2 1 C1 Three diagonals correctly drawn C1 23 6 19 2 15 10 18 1 14 22 17 5 13 21 9 4 12 25 8 16 11 24 7 20 3 3 B2 for the 11 numbers on the diagonals correctly placed (excluding 19) or B1 for 13 in the middle 5(b)(ii) 65 ………. 5 2 B1 for each

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2024 © Cambridge University Press & Assessment 2024 Page 7 of 7 Question Answer Marks Partial Marks 6 [2,] 5, 8, 11, 14, 17, 20, 23, 26 C1 Four of these correct 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 2 correct or B1 for 1 correct If 0 scored excluding communication, SC2 for 2 squares correctly reflected from their first square or SC1 for 1 square correctly reflected from their first square