Cambridge IGCSE Mathematics - International 0607 — 2021 Oct/Nov Paper 5 · Variant 1

This document has 8 pages. Any blank pages are indicated. Cambridge IGCSE™ DC (PQ) 212407/1 © UCLES 2021 [Turn over * 1 3 6 4 7 1 1 2 5 5 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 Investigation (Core) October/November 2021 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 0607/51/O/N/21 © UCLES 2021 Answer all the questions. ADDING SQUARE NUMBERS This investigation looks at adding two or more square numbers to make another square number. In this investigation all numbers are positive integers. 1 Complete the list of the first six square numbers. 1 1 2 = 22 = … 3 9 2 = 42 = … 52 = … 6 36 2 = [1] 2 (a) Work out (i) 92, … [1] (ii) 402. … [1] (b) Show that 9 40 41 2 2 2 + = . [2] 3 When a b c 2 2 2 + = then (a, b, c) is a 3-square set. a, b and c are positive integers. Example In Question 2(b), a 9 = , b 40 = and c 41 = . 9 40 41 2 2 2 + = , so (9, 40, 41) is a 3-square set. When a b c 2 2 2 + = then c a b 2 2 = + . Use this formula and any patterns you notice to complete the table on the next page for 3-square sets.

3 0607/51/O/N/21 © UCLES 2021 [Turn over a b c 3 4 5 5 12 13 7 24 25 9 40 41 11 60 13 84 85 112 113 144 19 181 21 221 25 312 313 [6] 4 When a b c d 2 2 2 2 + + = then (a, b, c, d) is a 4-square set. It is possible to make a 4-square set using two rows in the table. Example From the table row two 5 12 13 2 2 2 + = row six 13 84 85 2 2 2 + = Replace 132 in the second equation with 5 12 2 2 + from the first equation: . 5 12 84 85 2 2 2 2 + + = So (5, 12, 84, 85) is a 4-square set. Use the same method with rows from the table to find two more 4-square sets. (… , … , … , …) and (… , … , … , …) [3]

4 0607/51/O/N/21 © UCLES 2021 5 (a) Show that (6, 12, 12, 18) is a 4-square set. [2] (b) k is any positive integer greater than 1. If (ka, kb, kc, kd) is a 4-square set, then ka kb kc kd 2 2 2 2 + + = ` ` ` ` j j j j . Show that (a, b, c, d) must also be a 4-square set. [2]

5 0607/51/O/N/21 © UCLES 2021 [Turn over (c) The numbers in the 4-square set (6, 12, 12, 18) have common factors. (i) Find a common factor of 6, 12, 12 and 18 that is greater than 1. … [1] (ii) Use (6, 12, 12, 18) and part (b) to find a 4-square set where a, b, c and d do not have a common factor greater than 1. (… , … , … , …) [2]

6 0607/51/O/N/21 © UCLES 2021 6 Here is another method for finding a 4-square set (a, b, c, d). Choose two positive integers a and b with a less than b. Then c a b 2 1 2 2 = + - and a b d 2 1 2 2 = + + make the 4-square set (a, b, c, d). (a) Use this to find a 4-square set when (i) a 2 = and b 3 = , (2, 3, … , …) [3] (ii) a 2 = and d 3 4 = . (2, … , … , 43) [3] (b) (i) Use your answers to part (a) and any patterns you notice to complete the table for 4-square sets that start with 2. a b c d 2 3 2 5 14 15 2 7 26 27 2 43 2 [3] (ii) Write down an equation connecting c and d. … [1]

7 0607/51/O/N/21 © UCLES 2021 (c) When a and b are both even then c a b 2 1 2 2 = + - and a b d 2 1 2 2 = + + do not give a 4-square set. Give an example to show this. [2] (d) When a and b are both odd there are no 4-square sets. In a 4-square set, d 23 = . (i) Show that a b 45 2 2 + = . [1] (ii) Find a 4-square set when d 23 = . (… , … , … , 23) [2]

8 0607/51/O/N/21 © UCLES 2021 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 2021 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) October/November 2021 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 October/November 2021 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2021 © UCLES 2021 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 October/November 2021 © UCLES 2021 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 October/November 2021 © UCLES 2021 Page 4 of 6 Question Answer Marks Partial Marks 1 4 16 25 1 2(a)(i) 81 1 2(a)(ii) 1600 1 2(b) 81 + 1600 = 1681 = 412 2 B1 for 1681 seen 3 At least three correct differences in at least one column or c = 2 2 + a b or a2 + b2 = c2 used numerically at least once C1 a b c 3 4 5 5 12 13 7 24 25 9 40 41 11 60 61 13 84 85 15 112 113 17 144 145 19 180 181 21 220 221 23 264 265 25 312 313 5 B1 for first column In middle and last columns B4 for all 6 cells correct, or B3 for 4 or 5 cells correct, or B2 for 2 or 3 cells correct, or B1 for 1 cell correct. 4 Using (3, 4, 5) and (5, 12, 13) or (7, 24, 25) and (25, 312, 313) C1 (3, 4, 12, 13) (7, 24, 312, 313) 2 B1 for each 4-square set If 0 scored, SC1 for writing squares 5(a) 62 + 122 + 122 = 324 nfww and 182 = 324 2 B1 for each statement

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2021 © UCLES 2021 Page 5 of 6 Question Answer Marks Partial Marks 5(b) k2a2 + k2b2 + k2c2 = k2d2 1 If 0 scored, SC1 for full reasons if k is numerical and greater than 1. Evidence of dividing by k2 or k2(a2 + b2 + c2) seen leading to a2 + b2 + c2 = d 2 1 5(c)(i) One from 2, 3 or 6 1 5(c)(ii) Divide by 6 oe C1 If 0 scored SC1 for (2, 4, 4, 6) or (3, 6, 6, 9) (1, 2, 2, 3) 1 6(a)(i) 2 2 2 3 1 2 + − or 2 2 2 3 1 2 + + C1 (2, 3, 6, 7) 2 B1 for 6 or 7 6(a)(ii) 2 2 2 1 43 2 + + = b C1 (2, 9, 42, 43) 2 B1 for 9 or 42 6(b)(i) At least 3 correct differences in column b, c or d or 2 2 2 11 1 2 + − or 2 2 2 11 1 2 + + C1 a b c d 2 3 6 7 2 5 14 15 2 7 26 27 2 9 42 43 2 11 62 63 2 B1 for one correct cell If 0 scored, SC1 for last cell one more than the second last 6(b)(ii) d = c + 1 oe 1 6(c) Substituting a and b both even in 2 2 1 2 + − a b or 2 2 1 2 + + a b M1 A correct numerical value and ‘not an integer’ oe A1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2021 © UCLES 2021 Page 6 of 6 Question Answer Marks Partial Marks 6(d)(i) 2 2 1 2 + + a b = 23 and a2 + b2 + 1 = 46 or 23 × 2 – 1 = 45 OR c = 22 and 2 2 1 2 + − a b = 2 and a2 + b2 – 1 = 44 or 22 × 2 + 1 = 45 1 6(d)(ii) (3, 6, 22, 23) 2 B1 for 3 and 6 or for 22