Cambridge IGCSE Mathematics - International 0607 — 2016 Oct/Nov Paper 5 · Variant 1

This document consists of 7 printed pages and 1 blank page. DC (CW/SG) 117233/1 © UCLES 2016 [Turn over * 8 4 5 3 9 7 5 0 6 9 * Cambridge inTernaTional maThemaTiCs 0607/51 Paper 5 (Core) october/november 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 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. Cambridge international examinations Cambridge International General Certificate of Secondary Education

2 0607/51/O/N/16 © UCLES 2016 Answer all the questions. INVESTIGATION SQUARES ON GRIDS This investigation looks at the number of squares drawn on square grids. All squares are drawn using gridlines. 1 Here is a 2 by 2 grid. Explain why there are 5 squares on a 2 by 2 grid. … … 2 Here is a 3 by 3 grid. Complete these statements about the numbers of different sized squares on a 3 by 3 grid. The number of 1 by 1 squares on a 3 by 3 grid is … The number of 2 by 2 squares on a 3 by 3 grid is 4 The number of 3 by 3 squares on a 3 by 3 grid is … So the total number of squares on a 3 by 3 grid is …

3 0607/51/O/N/16 © UCLES 2016 [Turn over 3 Complete these statements about the numbers of different sized squares on a 4 by 4 grid. You may use the grids below to help you. The number of 1 by 1 squares on a 4 by 4 grid is … The number of 2 by 2 squares on a 4 by 4 grid is … The number of 3 by 3 squares on a 4 by 4 grid is … The number of 4 by 4 squares on a 4 by 4 grid is … So the total number of squares on a 4 by 4 grid is 30

4 0607/51/O/N/16 © UCLES 2016 4 (a) Use your results from questions 1, 2 and 3 to help you complete this table. Size of grid Number of…. Total number of squares 1 by 1 squares 2 by 2 squares 3 by 3 squares 4 by 4 squares 5 by 5 squares 6 by 6 squares 1 by 1 1 1 2 by 2 5 3 by 3 4 4 by 4 30 5 by 5 6 by 6 (b) What is the mathematical name for the numbers in the 1 by 1 squares column? … (c) Work out the total number of squares on an 8 by 8 grid. … (d) Here is part of a table for an n by n grid. It only has columns for 1 by 1 squares up to 6 by 6 squares. Complete the table using expressions in terms of n. Size of grid Number of…. 1 by 1 squares 2 by 2 squares 3 by 3 squares 4 by 4 squares 5 by 5 squares 6 by 6 squares n by n (n – 4)2 (e) Write an expression, in terms of n, for the number of 12 by 12 squares on an n by n grid. …

5 0607/51/O/N/16 © UCLES 2016 [Turn over (f) (i) Find the number of 5 by 5 squares on a 20 by 20 grid. … (ii) The number of 5 by 5 squares on an n by n grid is 36. Find the value of n. … 5 Here is a formula for the total number of squares, T, on an n by n grid. T n n n d 3 2 6 3 2 = + + + (a) The total number of squares on a 1 by 1 grid is 1. Show that d = 0. (b) Show that the formula gives the correct total number of squares on a 4 by 4 grid. (c) Find the total number of squares on a 10 by 10 grid. …

6 0607/51/O/N/16 © UCLES 2016 6 (a) There are nine 7 by 7 squares on a 9 by 9 grid. The diagrams show a 7 by 7 square drawn in two positions on a 9 by 9 grid. In each diagram the same 2 by 2 square is shaded. Consider the possible positions of the 7 by 7 square. Explain how the shaded 2 by 2 square can be used to calculate the number of 7 by 7 squares on a 9 by 9 grid. … … …

7 0607/51/O/N/16 © UCLES 2016 (b) A square which is bigger than 9 by 9 is drawn on a square grid. It is only possible to draw 25 of these squares on the square grid. Find two possible sizes for the square and the grid it is drawn on. You may use the grid below to help you. Square size … by … on a … by … grid. Square size … by … on a … by … grid.

8 0607/51/O/N/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 0607/51 Paper 5 (Core) October/November 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 October/November 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 – October/November 2016 0607 51 © UCLES 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 4 small and 1 large oe 1 2 9 4 1 14 2 B1 for 9 and 1 B1FT for sum of their 1, 4 and their 9 3 16 9 4 1 30 2 B1 for either 9 or 4 If reverse order in question 2 then SC2 for reverse order 4 (a) Size Total 1 by 1 1 1 2 by 2 4 1 5 3 by 3 9 4 1 14 4 by 4 16 9 4 1 30 5 by 5 25 16 9 4 1 55 6 by 6 36 25 16 9 4 1 91 3 B1 for rows 2 to 4 correct B1 for row 5 correct B1 for row 6 correct If 0 scored SC1 for one of columns 2, 3 or 4 correct If reverse order in question 2 then SC1 for rows 2 to 4 with reverse sequence SC1 for row 5 with reverse sequence SC1 for row 6 with reverse sequence (b) Square [numbers] 1 (c) 204 2 B1 for 49 and 64 seen C opportunity (d) n2 (n – 1)2 (n – 2)2 (n – 3)2 … (n – 5)2 2 B1 for 2 correct (e) (n – 11)2 oe 1 (f) (i) 256 1 C opportunity (ii) 10 1 C opportunity

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 51 © UCLES 2016 Question Answer Mark Part Marks 5 (a) 1 = 1 1 1 3 2 6 + + + d so d = 0 oe 1 (b) 3 2 4 4 4 30 3 2 6 + + = soi 1 (c) 385 2 M1 for 3 2 10 10 10 3 2 6 + + or 91 + 72 + 82 + 92 + 102 oe 6 (a) The upper right corner of the large square can be put on any of the nine points in the 2 by 2 square oe or Use the 5 surrounding squares and the 4 squares inside the 2 by 2 square oe 1 (b) Two from: 10 by 10 on 14 by 14 11 by 11 on 15 by 15 12 by 12 on 16 by 16 etc. 2 B1 for one C opportunity Communication: Seen in two of the following questions 1 4 (c) For showing 91 + 49 + 64 or 1 + 4 + 9 + 16 + 25 + 36 + 49 + 64 or tabular form (without plus signs) 4 (f) (i) For (n – 4)2 or their (20 – 4)2 oe 4 (f) (ii) For (n – 4)2 = 36 or (10 – 4)2 = 36 or √36 = 6 or 6 + 4 = 10 or 8 9 10 16 25 36 or for a 10 by 10 grid there are 100, 81, 64, 49, 36 squares 6 (b) For a square of side 4 or 4 by 4 seen or used