Cambridge IGCSE Mathematics - International 0607 — 2016 May/June Paper 5 · Variant 2

This document consists of 6 printed pages and 2 blank pages. DC (NF) 115859/2 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 1 4 6 5 5 1 2 7 6 1 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 Paper 5 (Core) May/June 2016 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 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 0607/52/M/J/16 © UCLES 2016 Answer all the questions. INVESTIGATION SUMS OF CONSECUTIVE INTEGERS This investigation looks at the results when the terms of a sequence of consecutive positive integers are added together. 1 Here are four sequences of consecutive positive integers. The sequence 5, 6, 7, 8, 9, 10, 11 has 7 terms. The median (the middle term) is 8. The sequence 7, 8 has only 2 terms. The median is 7.5 . The sequence 20, 21, 22, 23, 24, 25 has 6 terms. The median is 22.5 . The sequence 20, 21, 22, …, 40 has 21 terms. The median is 30. For a sequence of consecutive integers, (a) give an example to show that the number of terms is calculated using the rule last term – first term + 1 (b) describe how to calculate the median using only the first term and the last term. … …

3 0607/52/M/J/16 © UCLES 2016 [Turn over 2 (a) Complete the table of sequences of consecutive positive integers. Sequence Number of terms Median Sum of all the terms 3, 4, 5, 6, 7, 8, 9 7 6 7, 8 2 7.5 20, 21, 22, … , 40 21 30 630 5, 6, 7 18 2, 3, 4, 5, 6, 7, 8, 9 8 6 4.5 27 5 7 (b) Explain how to calculate the sum of all the terms using only the number of terms and the median. … (c) What is always true about the number of terms when the median is an integer? … (d) What is always true about the median when the number of terms is even? …

4 0607/52/M/J/16 © UCLES 2016 3 Use your answer to question 2(b) to help you complete the table of sequences of two or more consecutive positive integers. Sequence Number of terms Median Sum 5 15 4 34 49 4 Use your answers to question 1 and question 2(b) to help you find the sum of this sequence. 15, 16, 17, …, 985. …

5 0607/52/M/J/16 © UCLES 2016 [Turn over 5 Sequences have 2 or more terms. Find all the sequences of consecutive positive integers that have a sum of 77.

6 0607/52/M/J/16 © UCLES 2016 6 (a) Use the factors of 16 to show why the sum of a sequence of consecutive positive integers cannot equal 16. (b) Find a number larger than 20 that cannot be written as the sum of consecutive positive integers. …

7 0607/52/M/J/16 © UCLES 2016 BLANK PAGE

8 0607/52/M/J/16 © UCLES 2016 BLANK PAGE 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.

® IGCSE is the registered trademark of Cambridge International Examinations. This document consists of 3 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 Paper 5 (Core) May/June 2016 MARK SCHEME Maximum Mark: 24 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 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 – May/June 2016 0607 52 © Cambridge International Examinations 2016 Abbreviations awrt answers which round to 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) An example and illustration of last – first + 1 1 (b) first+last 2 oe 1 2 (a) Sequence Median Sum 3, 4, 5, 6, 7, 8, 9 7 6 42 7, 8 2 7.5 15 20, 21, 22, …, 40 21 30 630 5, 6, 7 3 6 18 2, 3, 4, 5, 6, 7, 8, 9 8 5.5 44 2, 3, 4, 5, 6, 7 6 4.5 27 5, 6, 7, 8, 9 5 7 35 5 B1 for first two rows B1 for each of the last four rows C opportunity (b) multiply 1 (c) odd [number] 1 (d) ends in .5 oe 1 3 Sequence Median Sum 4, 5, 6 3 5 15 7, 8, 9, 10 4 8.5 34 4, 5, 6, 7, 8, 9, 10 or 24, 25 7 or 2 7 or 24.5 49 5 1st row: B1 2nd row:B1 for sequence B1 for median 3rd row: B1 for sequence B1 for median

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 52 © Cambridge International Examinations 2016 Question Answer Mark Part Marks 4 485 500 2 B1 500 or 971 seen C opportunity 5 38, 39 8, 9, 10, 11, 12, 13, 14 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12 3 B1 for each sequence or B1 for 2 × 38.5 B1 for 7 × 11 or 11 × 7 6 (a) 1, 2, 4, 8, 16 oe median not an integer when even number of terms oe 1 1 (b) 32 or 64 or 128, etc 1 Communication seen in one of 2(a), 3, 4, 5 or 6(b) 1