Cambridge IGCSE Mathematics - International 0607 — 2018 Oct/Nov Paper 5 · Variant 1

* 0 7 2 6 1 3 4 0 8 0 * This document consists of 8 printed pages. DC (RW/CGW) 153616/1 © UCLES 2018 [Turn over CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) October/November 2018 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/18 © UCLES 2018 Answer all the questions. INVESTIGATION DOTS IN RECTANGLES This investigation looks at the number of dots inside rectangles drawn on square dotty paper. 1 Rectangles with a height of 1 unit have no dots inside. height 1 width 1 height 1 width 2 height 1 width 3 height 1 width 4 (a) (i) These rectangles all have a height of 2 units. height 2 width 1 height 2 width 2 height 2 width 3 height 2 width 4 Complete the table. Height of rectangle (h) Width of rectangle (w) Number of dots inside rectangle (d) 2 1 2 2 2 3 2 4 (ii) Find an expression, in terms of w, for the number of dots, d, inside rectangles of height 2. …

3 0607/51/O/N/18 © UCLES 2018 [Turn over (b) Complete the tables below. You may use the square dotty paper on this page to help you. Height of rectangle (h) Width of rectangle (w) Number of dots inside rectangle (d) 3 1 3 2 3 3 3 4 3 w 2(w - 1) Height of rectangle (h) Width of rectangle (w) Number of dots inside rectangle (d) 4 1 4 2 4 3 4 4 4 w

4 0607/51/O/N/18 © UCLES 2018 (c) (i) Find a formula, in terms of h and w, for the number of dots, d, inside a rectangle of height h and width w. d = … (ii) Use your formula in part (i) to show that there are 30 dots inside a rectangle of height 6 and width 7. (iii) There are 33 dots inside a rectangle. Find the height and the width of this rectangle. … 2 (a) Squares are formed when h = w. Change your formula in question 1(c)(i) to find the number of dots, d, inside a square of side s. … (b) Find the number of dots inside a square of side 10. …

5 0607/51/O/N/18 © UCLES 2018 [Turn over 3 The rectangles below are drawn at an angle of 45° to the horizontal. They are called diagonal rectangles. The diagonal lines between the dots give the height and the width of the diagonal rectangles. width 3 width 2 width 1 height 1 height 1 height 1 height 2 width 1

6 0607/51/O/N/18 © UCLES 2018 (a) Complete the tables below. Height of diagonal rectangle (h) Width of diagonal rectangle (w) Number of dots inside diagonal rectangle (d) 1 1 1 1 2 2 1 3 3 1 4 Height of diagonal rectangle (h) Width of diagonal rectangle (w) Number of dots inside diagonal rectangle (d) 2 1 2 2 2 2 3 2 4 2 w 3w - 1 Height of diagonal rectangle (h) Width of diagonal rectangle (w) Number of dots inside diagonal rectangle (d) 3 1 3 2 3 3 3 4

7 0607/51/O/N/18 © UCLES 2018 [Turn over (b) Use your results from part (a) and any patterns you notice to complete the following table with expressions, in terms of the width w, for the number of dots inside diagonal rectangles. Height of diagonal rectangle (h) Number of dots inside diagonal rectangle (d) 1 2 3w - 1 3 4 5 9w - 4 (c) (i) Find a formula, in terms of h and w, for the number of dots, d, inside a diagonal rectangle. … (ii) Use your formula in part (i) to find the number of dots inside the diagonal rectangle of height 10 and width 3. … Question 4 is printed on the next page.

8 0607/51/O/N/18 © UCLES 2018 4 (a) Diagonal squares are formed when h = w. Change your formula in question 3(c)(i) to find the number of dots, d, inside a diagonal square of side s. … (b) There are 181 dots in a diagonal square. Find the length of one side. … 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.

This document consists of 6 printed pages. © UCLES 2018 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) October/November 2018 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 International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the October/November 2018 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 2 of 6 Generic Marking Principles These general marking principles must be applied by all examiners when marking candidate answers. They should be applied alongside the specific content of the mark scheme or generic level descriptors for a question. Each question paper and mark scheme will also comply with these marking principles. GENERIC MARKING PRINCIPLE 1: Marks must be awarded in line with: • the specific content of the mark scheme or the generic level descriptors for the question • the specific skills defined in the mark scheme or in the generic level descriptors for the question • the standard of response required by a candidate as exemplified by the standardisation scripts. GENERIC MARKING PRINCIPLE 2: Marks awarded are always whole marks (not half marks, or other fractions). GENERIC MARKING PRINCIPLE 3: Marks must be awarded positively: • marks are awarded for correct/valid answers, as defined in the mark scheme. However, credit is given for valid answers which go beyond the scope of the syllabus and mark scheme, referring to your Team Leader as appropriate • marks are awarded when candidates clearly demonstrate what they know and can do • marks are not deducted for errors • marks are not deducted for omissions • answers should only be judged on the quality of spelling, punctuation and grammar when these features are specifically assessed by the question as indicated by the mark scheme. The meaning, however, should be unambiguous. GENERIC MARKING PRINCIPLE 4: Rules must be applied consistently e.g. in situations where candidates have not followed instructions or in the application of generic level descriptors. GENERIC MARKING PRINCIPLE 5: Marks should be awarded using the full range of marks defined in the mark scheme for the question (however; the use of the full mark range may be limited according to the quality of the candidate responses seen). GENERIC MARKING PRINCIPLE 6: Marks awarded are based solely on the requirements as defined in the mark scheme. Marks should not be awarded with grade thresholds or grade descriptors in mind.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 3 of 6 MARK SCHEME NOTES The following notes are intended to aid interpretation of mark schemes in general, but individual mark schemes may include marks awarded for specific reasons outside the scope of these notes. Types of mark M Method marks, awarded for a valid method applied to the problem. A Accuracy mark, awarded for a correct answer or intermediate step correctly obtained. For accuracy marks to be given, the associated Method mark must be earned or implied. B Mark for a correct result or statement independent of Method marks. When a part of a question has two or more ‘method’ steps, the M marks are in principle independent unless the scheme specifically says otherwise; and similarly where there are several B marks allocated. The notation ‘dep’ is used to indicate that a particular M or B mark is dependent on an earlier mark in the scheme. Abbreviations awrt answers which round to cao correct answer only dep dependent FT follow through after error isw ignore subsequent working nfww not from wrong working oe or equivalent rot rounded or truncated SC Special Case soi seen or implied

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 4 of 6 INVESTIGATION DOTS IN RECTANGLES Question Answer Marks Partial Marks 1(a)(i) 0 1 2 3 1 1(a)(ii) w – 1 oe 1 1(b) 0 2 4 6 2(w − 1) 0 3 6 9 3(w − 1) oe 3 B1 for 0, 2, 4 and 6 B1 for 0, 3, 6 and 9 B1 for 3(w – 1) oe C opportunity 1(c)(i) (h − 1)(w − 1) oe 2 B1 for first bracket B1 for second bracket C opportunity 1(c)(ii) (6 − 1)(7 − 1) oe 1 Dependent on (c)(i) 1(c)(iii) 4 [and] 12 OR 2 [and] 34 1 C opportunity 2(a) d = (s − 1)(s − 1) oe 1 FT their 1(c)(i) with both h and w 2(b) 81 1 FT their (a) C opportunity

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 5 of 6 Question Answer Marks Partial Marks 3(a) 1 2 3 4 2 5 8 11 3 8 13 18 2 B1 for first two tables B1 for third table C opportunity 3(b) w 5w − 2 7w – 3 oe 3 B1 for each C opportunity 3(c)(i) d = (2h − 1)w − (h − 1) oe 2 B1 for (2h − 1) × by w must be implied B1 for (h − 1) for the numerical term must be implied oe C opportunity 3(c)(ii) 48 1 FT their part (i) with both h and w C opportunity 4(a) d = (2s − 1)s − (s − 1) oe 1 FT their 3(c)(i) with both h and w 4(b) 10 2 M1 for substitution of 181 for d in their 4(a) C opportunity

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 6 of 6 Question Answer Marks Partial Marks Communication: seen in six of the following questions 2 1 mark for three opportunities seen 1(b) at least 3 correct drawings of rectangles of height 3 or 4 and widths 1 to 4 or for differences of 3 seen 1(c)(i) 3 correct expressions showing difference of 1 in either coefficients or constants 1(c)(iii) 11 × 3 or 3 × 11 or 33 × 1 or 1 × 33 seen or 33 ÷ 3 = 11 etc. 2(b) Correct substitution into their formula in 2(a) 3(a) at least 3 drawings of different diagonal rectangles of height 2 or 3 and widths 1 to 4 or substitution seen for second table 3(b) At least two differences of 5 seen 3(c)(i) Showing differences of 2 in w coefficients or extending the table in 3(b). 3(c)(ii) Correct substitution into their formula in 3(c)(i) or into 19w – 9 by extending the table to h = 20. 4(b) Further working to solve their quadratic after M1 e.g. by sketch or at least 3 trials 2(a), 3(c)(i), 4(a) d = written in all three answers