Cambridge IGCSE Mathematics - International 0607 — 2024 May/June Paper 5 · Variant 2

This document has 8 pages. Any blank pages are indicated. [Turn over DC (JP) 332301/2 © UCLES 2024 * 4 8 3 6 7 2 0 2 8 8 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 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 [ ]. Cambridge IGCSE™ ,

 


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2 0607/52/M/J/24 © UCLES 2024 Answer all the questions. INVESTIGATION INTEGER TREES This investigation looks at patterns when integers are arranged in the shape of a tree. A tree uses consecutive integers starting with 1 at the top. Each tree has a trunk, branches and leaves. The integers always increase going away from the trunk, on both the right and the left. The first branch is always on the right. This tree uses the integers 1 to 6. It has 2 branches with 2 leaves on each branch. 1 2 3 6 5 4 branch trunk leaf Left Right This tree uses the integers 1 to 12. It has 4 branches with 2 leaves on each branch. The leaves furthest from the trunk are the outer leaves. 1 2 3 8 9 6 5 12 11 4 7 10 right outer leaves left outer leaves A is the sum of the left outer leaves. B is the sum of the right outer leaves. T is the sum of the integers in the trunk. Example For the tree with 4 branches, A 6 12 18 = + = B 3 9 12 = + = T 1 4 7 10 22 = + + + = . * 0019655401002 * ,

 


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3 0607/52/M/J/24 © UCLES 2024 [Turn over In Questions 1 to 5, the number of branches, n, is always even. 1 This tree has 6 branches with 2 leaves on each branch. 1 2 3 8 9 14 15 6 5 12 11 18 17 4 7 10 13 16 (a) Complete the calculation for A. A 6 12 18 = + + = … [1] (b) Find the value of B. … [2] (c) Find the value of T. … [2] 2 (a) Complete the table for trees with 2 leaves on each branch. Use your answers to Question 1 to help you. Number of branches (n) A B T A B - A B T + - 2 6 3 5 3 4 4 18 12 22 6 8 6 9 8 60 16 [4] (b) (i) Use values from the table to give an example to show that . A B n 1 5 = - . … [1] (ii) Write an expression for A B T + - in terms of n. … [1] * 0019655401003 * ,

 


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4 0607/52/M/J/24 © UCLES 2024 3 In this question trees have 3 leaves on each branch. This tree has 6 branches. 1 2 3 10 11 18 19 4 12 20 7 6 15 14 23 22 8 16 24 5 9 13 17 21 (a) Complete the table. Number of branches (n) A B T A B - A B T + - 2 8 4 6 4 6 4 24 16 28 8 12 6 66 18 8 80 [4] (b) (i) Write an expression for A B - in terms of n. … [1] (ii) Write an expression for A B T + - in terms of n. … [1] * 0019655401004 * ,

 


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5 0607/52/M/J/24 © UCLES 2024 [Turn over 4 In this question trees have 4 leaves on each branch. The table shows values of A B - and A B T + - . Complete the last row of the table with expressions in terms of n. Number of branches (n) A B - A B T + - 2 5 8 4 10 16 6 15 24 8 20 32 n [2] 5 (a) Complete the table. Use your answers from Questions 2(b)(ii), 3(b) and 4 to help you. Number of branches Number of leaves on each branch (x) A B - A B T + - n 2 1.5n n 3 n 4 n 5 [2] (b) A tree has an even number of branches, n, with x leaves on each branch. Complete the expressions for A B - and A B T + - in terms of n and x. A B - = n 2 ( …) A B T + - = … [2] (c) A tree has 16 branches and 7 leaves on each branch. Show that A B 64 - = and A B T 112 + - = . [2] * 0019655401105 * ,

 

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6 0607/52/M/J/24 © UCLES 2024 6 In this question trees have an odd number of branches. Trees have more than one branch. This tree has 5 branches with 2 leaves on each branch. 1 2 3 8 9 14 15 6 5 12 11 4 7 10 13 (a) Find the value of A B - and the value of A B T + - for this tree. A B - = … A B T + - = … [4] (b) Show that your expression for A B - in Question 5(b) does not give the correct value for this tree. [2] (c) Show that your expression for A B T + - in Question 5(b) gives the correct value for this tree. [1] * 0019655401106 * ,

 

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7 0607/52/M/J/24 © UCLES 2024 (d) There are different trees with odd numbers of branches where A B T 18 + - = . Find the number of branches and the number of leaves on each branch for all of these trees. … [4] * 0019655401107 * ,

 

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8 0607/52/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 * 0019655401108 * ,

 

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This document consists of 5 printed pages. © Cambridge University Press & Assessment 2024 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 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.

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2024 © Cambridge University Press & Assessment 2024 Page 2 of 5 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 May/June 2024 © Cambridge University Press & Assessment 2024 Page 3 of 5 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 May/June 2024 © Cambridge University Press & Assessment 2024 Page 4 of 5 Question Answer Marks Partial Marks 1(a) 36 1 1(b) 3 + 9 + 15 C1 27 1 1(c) 1 + 4 + 7 + 10 + 13 + 16 C1 51 1 2(a) 19 20 21 24 23 22 or 3 common differences of 3 or 4 seen in one of last two columns or 36 + 27 – 51 [= 12] or 60 – 12 = 48 C1 n A B T A – B A + B – T 2 6 3 5 3 4 4 18 12 22 6 8 6 36 27 51 9 12 8 60 48 92 12 16 3 48, 92, 12, 12 B2 for three correct or B1 for one correct 2(b)(i) 3 = 1.5 × 2 or 6 = 1.5 × 4 or 9 = 1.5 × 6 or 12 = 1.5 × 8 1 2(b)(ii) 2n oe 1 3(a) 25 26 27 28 32 31 30 29 or 3 common differences of 4 or 6 seen in one of last two columns C1 n A B T A – B A + B – T 2 8 4 6 4 6 4 24 16 28 8 12 6 48 36 66 12 18 8 80 64 120 16 24 3 B2 for five correct or B1 for three correct 3(b)(i) 2n oe 1 3(b)(ii) 3n oe 1 4 2.5n oe and 4n oe 2 B1 for each

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2024 © Cambridge University Press & Assessment 2024 Page 5 of 5 Question Answer Marks Partial Marks 5(a) branches leaves A – B A + B – T n 2 1.5n 2n n 3 2n 3n n 4 2.5n 4n n 5 3n oe 5n oe 2 B1 for each 5(b) x + 1 and xn oe 2 B1 for each 5(c) A – B =   16 7 1 2  or   16 7 1 2  = 64 and A + B – T = 7 × 16 or 7 × 16 = 112 2 B1 for each 6(a) A = 18 and B = 27 and T = 35 OR A + B – T = 18 + 27 – 35 or A – B = 18 – 27 and A + B – T = 45 – 35 C2 C1 for two correct of A, B and T OR 18 + 27 – 35 oe or 18 – 27 oe and 45 – 35 oe –9 1 10 1 6(b) Substitution of x = 2 and n = 5 into their expression 1 FT their x + 1 7.5 nfww 1 FT their 5 2 (x + 1) 6(c) 2 × 5 = 10 1 6(d) xn = 18 or tree drawn with odd number of branches and x × n = 18 C1 FT their xn list at least two from 1, 18 2, 9 3, 6 C1 3 branches 6 leaves 9 branches 2 leaves 2 B1 for each