Cambridge IGCSE Mathematics - International 0607 — 2016 Oct/Nov Paper 5 · Variant 2

This document consists of 7 printed pages and 1 blank page. DC (CW/SG) 117490/2 © UCLES 2016 [Turn over * 2 2 3 9 0 9 7 8 8 7 * Cambridge inTernaTional maThemaTiCs 0607/52 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 0607/52/O/N/16 © UCLES 2016 Answer all the questions. INVESTIGATION RECTANGLES WITHIN RECTANGLES This investigation looks for a method to find the number of rectangles when you draw horizontal and vertical lines inside a rectangle. One horizontal line, AB, is drawn inside a rectangle PQRS. The total number of rectangles is 3. They are PQBA, PQRS and ABRS. P A S Q B R 1 (a) Another line CD is drawn inside the rectangle PQRS. The total number of rectangles is now 6. P A C S Q B D R Four of the 6 rectangles are PQBA, PQDC, PQRS and ABDC. Complete the table to show the other two rectangles. PQBA PQDC PQRS ABDC

3 0607/52/O/N/16 © UCLES 2016 [Turn over (b) Three horizontal lines, AB, CD and EF are drawn inside the rectangle PQRS. P A C E S Q B D F R Complete the table to show all ten rectangles. PQBA PQRS ABDC ABRS CDRS (c) Four horizontal lines are drawn inside the rectangle. Find the total number of rectangles. …

4 0607/52/O/N/16 © UCLES 2016 (d) Complete the table. Number of horizontal lines inside the rectangle 0 1 2 3 4 5 6 7 Total number of rectangles 3 6 10 36 (e) The numbers in the bottom row of the table in part (d) form a sequence. Write down the mathematical name of these numbers. … (f) Ten horizontal lines are drawn inside the rectangle. Find the total number of rectangles. …

5 0607/52/O/N/16 © UCLES 2016 [Turn over 2 One vertical line, AB, is drawn inside rectangle PQRS. The total number of rectangles is 3. They are PABS, PQRS and AQRB. P A Q S B R (a) Two vertical lines are drawn inside a rectangle. Find the total number of rectangles. … (b) Complete the table. Number of vertical lines inside a rectangle 0 1 2 3 4 5 6 7 Total number of rectangles 3 (c) What is the connection between the table in question 1(d) and the table in question 2(b)? …

6 0607/52/O/N/16 © UCLES 2016 3 12 vertical lines are drawn inside a rectangle. Show that the total number of rectangles is given by the calculation 2 12 3 12 2 2 # + + . 4 (a) When n vertical lines are drawn inside a rectangle the total number of rectangles, T, is T n an b 2 1 2 = + + , where a and b are constants. Find the value of a and the value of b. Use your answers to write down the formula for T. a = … b = … T = …

7 0607/52/O/N/16 © UCLES 2016 (b) Use your formula in part (a) to show that when 7 vertical lines are drawn inside a rectangle, the number of rectangles is 36. (c) Calculate how many vertical lines are drawn when there are 231 rectangles. … 5 When 30 horizontal lines are drawn inside a rectangle, find the total number of rectangles. …

8 0607/52/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 0607/52 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 52 © 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 Marks Part Marks 1 (a) PQBA PQDC PQRS ABDC ABRS CDRS 2 B1 for each (b) PQBA PQDC PQFE PQRS ABDC ABFE ABRS CDFE CDRS EFRS 3 B2 for 3 or 4 correct or B1 for 2 correct (c) 15 1 C opportunity (d) Number of lines 0 1 2 3 4 5 6 7 Number of rectangles 1 3 6 10 15 21 28 36 3 B1 each cell C opportunity (e) Triangle [numbers] 1 (f) 66 1 C opportunity 2 (a) 6 1 (b) Number of lines 0 1 2 3 4 5 6 7 Number of rectangles 1 3 6 10 15 21 28 36 1 Allow one error (c) same 1 3 91 shown as answer to calculation 91 shown as 13th term in the sequence oe 1 1

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – October/November 2016 0607 52 © UCLES 2016 Question Answer Marks Part Marks 4 (a) [a=] 3 2 oe [b=] 1 3 B2 for either a or b correct If 0 scored SC2 for 2 3 2 2 + + n n seen or M1 for one correct substitution of T and n C opportunity (b) Substitution of 7 in their formula 1 FT (c) 20 2 M1 for n2 + 3n + 2 = 462 or for sketch or for correct sequence to 15th term or further 5 496 1 FT from their formula in 4(a) C opportunity Communication: Seen in one of the following questions 1 1 (c) Method of counting (implied addition), e.g. drawing or 5 + 4 + 3 + 2 + 1 Or listing rectangles 1 (d) Differences shown 1 (f) Working shown, e.g. sequence continued – 45, 55, 66 4 (a) Working shown e.g. difference method or substitution to give two equations 5 Working shown e.g. substitution