Cambridge IGCSE Mathematics - International 0607 — 2024 May/June Paper 5 · Variant 1

This document has 8 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 Investigation (Core) May/June 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 5 6 9 7 5 7 3 9 5 * DC (PQ) 332289/2 © UCLES 2024

2 0607/51/M/J/24 © UCLES 2024 Answer all the questions. INVESTIGATION SUMS OF NUMBERS This investigation looks at the sum of the positive integers, 1 2 3 g + + + . It also looks at the sum of their cubes, 1 2 3 3 3 3 g + + + . 1 (a) Calculate 1 2 3 4 5 6 7 8 + + + + + + + . … [1] (b) The numbers can also be added using this method. Complete this method for the sum of the numbers in part (a). Step 1 Step 2 Step 3 Step 4 Write down the first half of the numbers in a row. Write down the second half of the numbers underneath the first half but in reverse order. Add each column of two numbers to make a third row. Find the total of the numbers in the third row by writing the calculation as a multiplication. 1 2 3 4 8 7 6 5 9 9 9 9 4 9 # = … [1] 2 Use the method in Question 1(b) to calculate the sum of the first six positive integers, 1 2 3 4 5 6 + + + + + . [4]

3 0607/51/M/J/24 © UCLES 2024 [Turn over 3 Complete the method to show that the sum of the first 30 positive integers is 465. So 1 2 3 28 29 30 465 g + + + + + + = . 1 2 3 g 15 … … … g 16 … … … g … … # … = 465 [3] 4 Find the sum of the first 160 positive integers, 1 2 3 160 g + + + + . … [3]

4 0607/51/M/J/24 © UCLES 2024 5 Use Questions 1, 2, 3 and 4 to help you complete the table. Number of positive integers starting at 1 Multiplication Sum 4 2 5 # 10 Question 2 6 Question 1 8 4 9 # 10 Question 3 30 465 50 25 51 # 1275 128 64 129 # Question 4 160 204 20 910 [4] 6 n 1 2 3 g + + + + has n positive integers and its sum is T. Find a formula for T in terms of n. … [3]

5 0607/51/M/J/24 © UCLES 2024 [Turn over 7 In the table in Question 5 the number of integers is always even. Does your formula in Question 6 give the correct total when n 7 = ? [3]

6 0607/51/M/J/24 © UCLES 2024 8 (a) Complete the table. Sum of first n positive integers Sum of first n cube numbers Calculation Sum (T ) Calculation Sum (S ) 1 2 + 3 1 2 3 3 + 1 8 + 9 1 2 3 + + 1 2 3 3 3 3 + + 36 1 2 3 4 + + + 1 8 27 64 + + + 15 1 2 3 4 5 3 3 3 3 3 + + + + 225 [4] (b) (i) Write down the mathematical name for the positive integers in the Sum (S ) column. … [1] (ii) Write a formula for S in terms of T. … [1]

7 0607/51/M/J/24 © UCLES 2024 (c) Use your answers to Question 6 and Question 8(b) to find the sum of the cubes of the first 40 positive integers. 1 2 3 40 3 3 3 3 g + + + + … [4] (d) n 1 2 3 396 900 3 3 3 3 g + + + + = Find how many cube numbers there are in this sum. … [4]

8 0607/51/M/J/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 6 printed pages. © Cambridge University Press & Assessment 2024 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) May/June 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 May/June 2024 series for most Cambridge IGCSE, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2024 © Cambridge University Press & Assessment 2024 Page 2 of 6 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.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2024 © Cambridge University Press & Assessment 2024 Page 3 of 6 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2024 © Cambridge University Press & Assessment 2024 Page 4 of 6 Question Answer Marks Partial Marks 1(a) 36 1 1(b) 36 1 2 1 2 3 6 5 4 7 7 7 3 × 7 = 21 seen 4 B1 for each row If 0 scored, SC1 for 21. 3 30 29 28 31 31 31 31 15 × 31 3 B1 for each row 4 80 × 161 C2 C1 for 80 or 161 seen relevantly 12 880 1 5 4 2 × 5 10 6 3 × 7 21 8 4 × 9 36 10 5 × 11 55 30 15 × 31 465 50 25 × 51 1275 128 64 ×129 8256 160 80 ×161 12 880 204 102 × 205 20 910 4 B1 for each cell 6 T = 2 n (n + 1) oe 3 B2 for 2 n (n + 1) oe or T = 2 n × n + 1 (i.e. no brackets) or B1 for 2 n × n + 1 (i.e. no brackets) or T = an expression in n when simplified

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2024 © Cambridge University Press & Assessment 2024 Page 5 of 6 Question Answer Marks Partial Marks 7 1 + 2 + 3 + … + 7 = 28 C1 7 2 (7 + 1) C1 FT their formula or expression Yes oe and 28 from the correct formula 1 8(a) 1+2 3 13+ 23 1 + 8 9 1+2+3 6 13+ 23 + 33 1+8+27 36 1+2+3+4 10 13+ 23 + 33 + 43 1+8+27+64 100 1+2+3+4+5 15 13+23+33+43+53 1+8+27+64+125 225 4 B3 for 5 cells or B2 for 3 cells or B1 for 1 cell 8(b)(i) Square numbers 1 8(b)(ii) S = T 2 1 8(c) Correct substitution of 40 in formula C1 FT their formula or expression Correct evaluation of next step C1 FT their substitution in their formula (their 820)2 C1 FT their formula in 8(b)(ii) 672 400 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2024 © Cambridge University Press & Assessment 2024 Page 6 of 6 Question Answer Marks Partial Marks 8(d) 396900 = 630 or 2 1 ( 1) 2 their n n        = 396 900 C1 Attempt to solve their 2 n (n + 1) = 630 or 2 1 ( 1) 2 their n n        = 396 900 by drawing a sketch of their 2 n (n + 1) or 2 1 ( 1) 2 their n n        and a horizontal line labelled 630 or 396 900 or their 2 n (n + 1) = 630 and n(n + 1) = 1260 and 1260 or by showing two correct trials for n between 20 and 50 C2 FT their 2 n (n + 1) if a quadratic FT their 630 C1 for a sketch of their 2 n (n + 1) or 2 1 ( 1) 2 their n n        or for their 2 n (n + 1) = 630 and n(n + 1) = 1260 or for one correct trial between 20 and 50 If 0 or 1 scored, SC1 for 1 + 2 + 3 + ………… + 35 = 630 If 0 scored, SC1 for 13 + 23 + 33 + ..… + 353 = 396900 35 1