Cambridge IGCSE Mathematics - International 0607 — 2015 Oct/Nov Paper 5 · Variant 3

This document consists of 7 printed pages and 1 blank page. DC (ST) 118176 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 7 5 5 5 9 0 1 6 8 5 * CambrIdgE InTErnaTIonal maThEmaTICs 0607/53 Paper 5 (Core) october/november 2015 1 hour Candidates answer on the Question Paper. Additional Materials: Graphics Calculator rEad ThEsE InsTrUCTIons FIrsT Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. Do not use staples, paper clips, glue or correction fluid. You may use an HB pencil for any diagrams or graphs. DO noT WRITE IN ANY BARCODES. Answer all the questions. You must show all relevant working to gain full marks for correct methods, including sketches. In this paper you will also be assessed on your ability to provide full reasons and to communicate your mathematics clearly and precisely. At the end of the examination, fasten all your work securely together. The total number of marks for this paper is 24.

2 0607/53/O/N/15 © UCLES 2015 THE INVESTIGATION STARTS ON PAGE 3.

3 0607/53/O/N/15 © UCLES 2015 [Turn over Answer all the questions. INVESTIGATION SUMS OF TWO SQUARES This investigation looks at the results when two square numbers are added together. 1 Here is a list of the first 11 prime numbers. 2 3 5 7 11 13 17 19 23 29 31 (a) In the list there are 4 numbers that are one more than a multiple of 4. These are called Pythagorean Primes. The smallest one is 5 and the largest one is 29. Write down the other two. 5, … , … , 29 (b) The 17th century French mathematician Albert Girard proved that every Pythagorean Prime equals the sum of two square numbers. Write your answers to part (a) as the sum of two square numbers. Two have been written down for you. 5 = 12 + 22 … = … + … … = … + … 29 = 22 + 52 (c) Another Pythagorean Prime is 101. Write 101 as the sum of two square numbers. 101 = … + …

4 0607/53/O/N/15 © UCLES 2015 2 The sum of two square numbers can equal another square number. For example, 32 + 42 = 9 + 16 = 25 = 52 We say that 3, 4, 5 is a Pythagorean Triple. (a) Show, by calculation, that 7, 24, 25 is a Pythagorean Triple. (b) Each row in this table is a Pythagorean Triple. Complete the table. Use patterns of numbers in the table to help you. 3 4 5 5 12 13 7 24 25 9 40 11 60 13 113

5 0607/53/O/N/15 © UCLES 2015 [Turn over (c) What is the connection between the square of the smallest number and the other two numbers in each Pythagorean Triple in the table? … … (d) Use your answer to part (c) and the patterns of numbers in the table to complete the following Pythagorean Triple. … , … , 421

6 0607/53/O/N/15 © UCLES 2015 3 , , x x x 2 1 1 - + is a Pythagorean Triple when x is a square number. (a) (i) Find the Pythagorean Triple when x = 16. … , … , … (ii) Check that your answer to part (a)(i) is a Pythagorean Triple. Use the method of the example in question 2. (b) In the table, x is the square of an even number. Each row is a Pythagorean Triple. x 2 x – 1 x + 1 (x = 16) (x = 36) 12 37 16 63 65 99 24 145 Write your answer to part (a)(i) in the first row of this table. Complete the three columns of the table. You may use patterns or the fact that , , x x x 2 1 1 - + is a Pythagorean Triple to help you.

7 0607/53/O/N/15 © UCLES 2015 (c) What is the connection between the square of the smallest number and the sum of the other two numbers in each of the Pythagorean Triples in the table? … … (d) Show algebraically that , , x x x 2 1 1 - + satisfies your connection in part (c).

8 0607/53/O/N/15 © UCLES 2015 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 International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. blANk PAGE

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the October/November 2015 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core), maximum raw mark 24 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the October/November 2015 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

Page 2 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 53 © Cambridge International Examinations 2015 Abbreviations cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent SC Special Case nfww not from wrong working soi seen or implied Question Answer Mark Part Marks 1 (a) 13 17 1 (b) 13 = 22 + 32 17 = 12 + 42 1 1 (c) 12 + 102 1 2 (a) 49 + 576 = 625 oe 2 B1 for two correct squares (b) 41 61 84 85 15 112 4 B1 for 15 B2 for second column (one for each cell) B1 for third column (c) equal to the sum oe 1 C opportunity (d) 29, 420 1 C opportunity 3 (a) (i) 8, 15, 17 1 (ii) 64 + 225 = 289 oe 2 B1 for one correct square (b) [8] [15] [17] 35 20 101 143 5 B2 for one correct cell B1 for each of the other three C opportunity (c) The square is twice the sum oe 1

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2015 0607 53 © Cambridge International Examinations 2015 Question Answer Mark Part Marks (d) (2 x )2 = 4x x – 1 + x + 1 = 2x 2 B1 for one statement seen or implied. Communication seen in one of 2(c), 2(d) or 3(b) 1