Cambridge IGCSE Mathematics - International 0607 — 2020 May/June Paper 5 · Variant 1

Cambridge IGCSE™ DC (LK) 187931/2 © UCLES 2020 [Turn over This document has 8 pages. Blank pages are indicated. CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 Investigation (Core) May/June 2020 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 [ ]. * 1 8 4 8 0 1 4 6 5 7 *

2 0607/51/M/J/20 © UCLES 2020 Answer all the questions. COMBINING TRIANGLE NUMBERS This investigation looks at results when adding or subtracting triangle numbers. Here is a table of the first 21 triangle numbers, T1 to T21. T1 T2 T3 T4 T5 T6 T7 T8 T9 T10 T11 T12 T13 T14 T15 T16 T17 T18 T19 T20 T21 1 3 6 10 15 21 28 36 45 55 66 78 91 105 120 136 153 171 190 210 231 1 Find the next two triangle numbers. T22 = … T23 = … [4] 2 (a) Complete the table. T1 1 T2 - T1 2 T3 - T2 T4 - T3 T5 - T4 T6 - T5 6 T T n n 1 - - [2] (b) (i) T T 100 n n 1 - = - . Write down the value of n. … [1] (ii) Write down the difference between the 50th and the 49th triangle numbers. … [1]

3 0607/51/M/J/20 © UCLES 2020 [Turn over 3 Complete the table for adding two consecutive triangle numbers. T1 1 T2 + T1 4 T3 + T2 9 T4 + T3 T5 + T4 T6 + T5 T T n n 1 + - [2] 4 (a) Use the last row of the table in Question 2(a) to complete the equation T T n n 1 - = - … Use the last row of the table in Question 3 to complete the equation T T n n 1 + = - … By adding these two equations together show that T n n 2 n 2 = + . [1] (b) Find T1000 . … [2]

4 0607/51/M/J/20 © UCLES 2020 5 (a) The table shows the difference of the squares of two consecutive triangle numbers. Complete the table. (T1)2 1 (T2)2 - (T1)2 8 (T3)2 - (T2)2 (T4)2 - (T3)2 (T5)2 - (T4)2 125 (T6)2 - (T5)2 216 (Tn)2 - (Tn-1)2 [3] (b) Calculate the difference between the squares of the 50th and the 49th triangle numbers. … [2] 6 The sum of two different triangle numbers sometimes equals another triangle number. When this happens, we have a triangle triple. Example • Start with the triangle number T 6 3 = . • From the table in question 2(a) T T 6 6 5 - = . So T T T 6 5 3 - = . • Rearrange the equation T T T 3 5 6 + = . • The triangle triple is then (3, 5, 6). The three different numbers must be written in order of increasing size. (a) Start with triangle number T 15 5 = and complete the method of the Example to find another triangle triple. T15 - … = … So … - … = T5 T5 + … = … The triangle triple is ( 5, … , … ) [4]

5 0607/51/M/J/20 © UCLES 2020 [Turn over (b) In the table, each row is a triangle triple. Use your answer to part (a) and any patterns you notice to complete the table. Triangle triple 3 5 6 4 9 10 5 6 7 [5] (c) Use the list of triangle numbers on page 2 to check the triangle triple beginning with 6. [1]

6 0607/51/M/J/20 © UCLES 2020 7 (a) The triangle numbers T1 and T3 are not consecutive. They are two apart. Complete the table for subtracting triangle numbers that are two apart. T3 - T1 5 T4 - T2 T5 - T3 T6 - T4 T7 - T5 13 T T n n 2 - - [4] (b) Use the triangle number T 45 9 = to find a triangle triple where • the smallest number is 9 • the difference between the other two numbers is 2. Hints: Use the last row of the table in part (a). Use a method similar to that in the Example in Question 6. ( 9 , … , … ) [4]

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8 0607/51/M/J/20 © UCLES 2020 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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge. BLANK PAGE

This document consists of 6 printed pages. © UCLES 2020 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) May/June 2020 MARK SCHEME Maximum Mark: 36 Published Students did not sit exam papers in the June 2020 series due to the Covid-19 global pandemic. This mark scheme is published to support teachers and students and should be read together with the question paper. It shows the requirements of the exam. The answer column of the mark scheme shows the proposed basis on which Examiners would award marks for this exam. Where appropriate, this column also provides the most likely acceptable alternative responses expected from students. Examiners usually review the mark scheme after they have seen student responses and update the mark scheme if appropriate. In the June series, Examiners were unable to consider the acceptability of alternative responses, as there were no student responses to consider. Mark schemes should usually be read together with the Principal Examiner Report for Teachers. However, because students did not sit exam papers, there is no Principal Examiner Report for Teachers for the June 2020 series. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the June 2020 series for most Cambridge IGCSE™ and Cambridge International A & AS Level components, and some Cambridge O Level components.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 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 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.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 3 of 6 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. 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 2020 © UCLES 2020 Page 4 of 6 Question Answer Marks Partial Marks 1 231 + 22 C1 253 B1 their 253 + 23 C1 276 B1 If 0 scored SC1 for at least three consecutive differences under the table 2(a) T1 1 T2 – T1 2 T3 – T2 3 T4 – T3 4 T5 – T4 5 T6 – T5 6 Tn – Tn–1 n 2 B1 for 3, 4 and 5 B1 for n 2(b)(i) 100 1 2(b)(ii) 50 1 3 T1 1 T2 + T1 4 T3 + T2 9 T4 + T3 16 T5 + T4 25 T6 + T5 36 Tn + Tn–1 n2 2 B1 for two of 16, 25 and 36 B1 for n2 4(a) 2Tn = n2 + n 1 4(b) 2 1000 1000 2 + C1 500 500 B1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 5 of 6 Question Answer Marks Partial Marks 5(a) (T1)2 1 (T2)2 – (T1)2 8 (T3)2 – (T2)2 27 (T4)2 – (T3)2 64 (T5)2 – (T4)2 125 (T6)2 – (T5)2 216 (Tn)2 – (Tn–1)2 n3 B2 B1 for 27 and 64 B1 for n3 oe Correct working for one value C1 62 – 32 or 102 – 62 or 36 – 9 or 100 – 36 5(b) 503 C1 FT substitution of 50 into their n3 if at least quadratic. or Substitution of 49 and 50 into 2 2 n n + and 12752 – 12252 125 000 B1 6(a) T15 – T14 = 15 T15 – T14 = T5 T5 + T14 = T15 (5, 14, 15) 4 B1 for each row 6(b) 3 5 6 4 9 10 5 14 15 6 20 21 7 27 28 B4 B1 for each cell Showing differences or using the method at least once or last column are triangle numbers C1 6(c) 21 + 210 = 231 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 6 of 6 Question Answer Marks Partial Marks 7(a) T3 – T1 5 T4 – T2 7 T5 – T3 9 T6 – T4 11 T7 – T5 13 Tn – Tn–2 2n – 1 B3 B1 for 7, 9 and 11 B2 for 2n – 1 oe or B1 for 2n + k oe seen Differences of 2 shown C1 7(b) their 2n – 1 = 45 or differences continued to 45 C1 276 – 231 = 45 seen or T23 – T21 = T9 C1 (9, 21, 23) B2 B1 for 21 or 23 seen on the answer line