Cambridge IGCSE Mathematics - International 0607 — 2015 May/June Paper 5 · Variant 3

This document consists of 7 printed pages and 1 blank page. DC (NH/CGW) 100444/4 © UCLES 2015 [Turn over * 2 2 5 9 2 3 1 1 8 1 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 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 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 0607/53/M/J/15 © UCLES 2015 Answer all the questions. INVESTIGATION T-VALUES A grid of any length and width 10 is numbered 1, 2, 3, … . 41 42 43 44 45 46 47 48 49 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 The grid has a letter T placed on it, as shown. The T has a horizontal bar of length 3 and a vertical bar of length 2. The T shown is shape 1 because the number in the top left square of the T is 1. T-values are found using this method. Method Calculation of T-value for shape 1 Square the number at the bottom of the T. 222 = 484 Multiply together the numbers at each end of the horizontal bar. 1 × 3 = 3 Take the second answer from the first answer to find the T-value. 484 – 3 = 481 The T-value for shape 1 is T1 = 481. This investigation is about finding T-values. 1 (a) Complete this table. Shape number n Working T-value Tn 1 222 – 1 × 3 = 484 – 3 T1 = 481 2 232 – 2 × 4 = 529 – 8 T2 = 521 3 T3 = 4 T4 = 5 T5 =

3 0607/53/M/J/15 © UCLES 2015 [Turn over (b) Explain how you know that T6 = 681 without using the method in the table. … … … (c) When a T is placed at the end of a line, it still has a T-value. The T “wraps round” like this. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Work out T9. T9 = …

4 0607/53/M/J/15 © UCLES 2015 2 The T is now placed on this 10 by 10 grid. 41 42 43 44 45 46 47 48 49 50 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 61 62 63 64 65 66 67 68 69 70 51 52 53 54 55 56 57 58 59 60 81 82 83 84 85 86 87 88 89 90 71 72 73 74 75 76 77 78 79 80 91 92 93 94 95 96 97 98 99 100 (a) Work out the greatest T-value for a T that fits completely on this grid. … (b) Complete this statement for the numbers in the grid. In each row the numbers increase by 1 and in each column the numbers increase by … .

5 0607/53/M/J/15 © UCLES 2015 [Turn over (c) Complete the squares in this T using expressions in terms of n. n n + 1 (d) Complete this working to show that Tn = 40n + 441. The first line of working has been started for you. Tn = ( n + … ) 2 – n ( n + … ) (e) When Tn = 2641, find the value of n. n = … (f) Explain why 840 cannot be a T-value. … …

6 0607/53/M/J/15 © UCLES 2015 3 The T is now placed on a new grid that is 11 squares wide. 45 46 47 48 49 50 51 52 53 54 1 2 3 4 5 6 7 8 9 10 12 13 14 15 16 17 18 19 20 21 23 24 25 26 27 28 29 30 31 32 34 35 36 37 38 39 40 41 42 43 55 11 22 33 44 T1, T2, T3, T4, T5, … form a sequence. (a) Complete this table. Shape number n 1 2 3 4 5 8 T-value Tn 573 661 705 (b) (i) Find a formula, in terms of n, for Tn . Tn = … (ii) Show that your formula in part b(i) gives the correct result for T10 .

7 0607/53/M/J/15 © UCLES 2015 4 A different T is placed on a grid that is w squares wide. The T has a horizontal bar of length 3 and a vertical bar of length 3. 1 2 3 4 5 6 7 8 9 10 w n Complete the squares in this T using expressions in terms of n and w. n

8 0607/53/M/J/15 © UCLES 2015 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. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the May/June 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 May/June 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 – May/June 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 1 (a) 561 601 641 1 1 1 If 0 scored SC1 for 242 – 3 × 5 252 – 4 × 6 all correct in working 262 – 5 × 7 (b) Increasing by 40 or 641 + 40 1 (c) 801 1 C opportunity 2 (a) 3561 2 M1 for their 992 – their 78 × their 80 (b) Ten or 10 1 (c) (top right) n + 2 oe (bottom) n + 21 oe 1 1 (d) [(n + 21)2 – n(n + 2)] n2 + 42n + 441 – n2 – 2n oe 2 B1 for n2 + 42n + 441 B1 for – n2 – 2n or B1 for 481, 521, 561, 601with differences 40, 40, 40 B1 dep for calculation to find 441 (e) 55 1 C opportunity (f) All T-results end in 1 oe [and this ends in 0 oe] or [n = ] 10.05 and n must be integer oe 1 3 (a) 617 749 881 2 B1 for one correct (b) (i) 44n + 529 2 B1 for 44n + k or jn + 529 C opportunity (ii) 44 × 10 + 529 = 969 and 33 × 33 – 10 × 12 = 969 1FT 1 FT their formula with n = 10

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 53 © Cambridge International Examinations 2015 4 [ n + 1 n + 2] n + w + 1 n + 2w + 1 n + 3w + 1 1 1FT 1FT FT their pattern adding only 10 each time Communication seen in one of 1(c), 2(e), 3(b)(i) 1