Cambridge IGCSE Mathematics - International 0607 — 2018 Oct/Nov Paper 5 · Variant 2

* 3 1 3 6 9 9 5 9 4 6 * This document consists of 7 printed pages and 1 blank page. DC (LK/CGW) 153617/1 © UCLES 2018 [Turn over CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 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 0607/52/O/N/18 © UCLES 2018 Answer all the questions. INVESTIGATION RIGHT SPIRALS This investigation is about the lengths of spirals drawn on a square grid. A robot starts from 0 and moves 1 unit to Corner 1. It then turns right and moves 1 unit to Corner 2. It then turns right and moves 2 units to Corner 3. It then turns right and moves 2 units to Corner 4. It then turns right and moves 3 units to Corner 5. This forms a spiral, shown on the grid below. 5 4 3 2 1 0 10 The robot continues to turn and move in the same way. 1 (a) Continue the spiral to Corner 10. (b) The length of the spiral from 0 to Corner 4 is 6 units. Find the length of the spiral from 0 to Corner 10. …

3 0607/52/O/N/18 © UCLES 2018 [Turn over (c) Use your spiral to complete this table. Corner number Lengths added Length from 0 1 1 1 2 1 + 1 2 3 1 + 1 + 2 4 4 6 5 6 1 + 1 + 2 + 2 + 3 + 3 12 7 1 + 1 + 2 + 2 + 3 + 3 + 4 16 8 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 9 10 1 + 1 + 2 + 2 + 3 + 3 + 4 + 4 + 5 + 5

4 0607/52/O/N/18 © UCLES 2018 2 This table shows the first five terms of a sequence. n 1 2 3 4 5 6 7 Term of the sequence 1 3 6 10 15 (a) For this sequence, fill in the next two terms. (b) Write down the mathematical name for this sequence of numbers. … (c) The nth term for this sequence is ( ) n n 2 1 + . Show that this is correct when n = 5. 3 This table shows the length, L, of the spiral from 0 to an even numbered corner, k. (a) Use your table from question 1(c) to help you complete this table. k Length (L) 2 2 4 6 6 12 8 10 12 14 56 16

5 0607/52/O/N/18 © UCLES 2018 [Turn over (b) Complete this table using your answers to question 2(a) and question 3(a). n Term of the sequence k Length (L) 1 1 2 2 2 3 4 6 3 6 6 12 4 10 8 5 15 10 6 12 (i) Complete this formula for n in terms of k. n = … (ii) Write down the connection between the length, L, and the term of the sequence. … (iii) Use part (i), part(ii) and question 2(c) to show that the formula for the length, L, of the spiral from 0 to an even numbered corner, k, is L k k 2 2 1 = + e o. (iv) Show that the formula from part(iii) is correct for Corner 6. (v) Show that the formula from part (iii) is not correct when k is an odd number.

6 0607/52/O/N/18 © UCLES 2018 4 (a) Write down the length of the spiral (i) from Corner 5 to Corner 6 , … (ii) from Corner 6 to Corner 7. … (b) When k is an even number, find an expression, in terms of k, for the length of the spiral (i) from Corner (k – 1) to Corner k, … (ii) from Corner k to Corner (k + 1). …

7 0607/52/O/N/18 © UCLES 2018 5 (a) Using question 3(b)(iii) and question 4 (b)(i), show that the length of the spiral from 0 to Corner 7 is 16 units. (b) Find the length of the spiral from 0 to Corner 91. …

8 0607/52/O/N/18 © UCLES 2018 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. BLANK PAGE

This document consists of 5 printed pages. © UCLES 2018 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 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.

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 2 of 5 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.

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 3 of 5 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

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 4 of 5 Question Answer Marks Partial Marks 1(a) Complete spiral to Corner 10 1 1(b) 30 1 C opportunity 1(c) 1 + 1 + 2 + 2 1 + 1 + 2 + 2 + 3 9 20 1 + 1+ 2 + 2 + 3 + 3 + 4 + 4 + 5 25 (30) 3 B1 for row 5 or row 9 correct B1 for 2 further correct cells 2(a) 21, 28 1 C opportunity 2(b) Triangle 1 Accept ‘triangular’ 2(c) 5(5 1) 15 2 + = 1 3(a) [their 20] [their 30] 42 72 1 C opportunity 3(b)(i) 2 k oe 1 3(b)(ii) L is double the term oe 1 Accept reverse and in words but must be clear what is doubled oe 3(b)(iii) 1 2 2 [ ] 2 2   +     = × k k L oe 1 Numerical examples are insufficient 3(b)(iv) 6 6 1 12 2 2   + =     1 3(b)(v) Substitute odd k into formula for L and work out the result correctly 1 State that the answer is not the number in the table oe or not a whole number oe or not at a spiral corner or not double the term 1 4(a)(i) 3 1 4(a)(ii) 4 1 4(b)(i) 2 k oe 1 C opportunity

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2018 © UCLES 2018 Page 5 of 5 Question Answer Marks Partial Marks 4(b)(ii) 1 2 + k oe 1 C opportunity 5(a) 6 6 1 2 2   +     or 8 8 1 2 2   +     or 7 7 1 2 2   +     1 May be in stages Substitution of 6, 7 or 8 must be seen. + 6 1 2 + or −8 2 or 1 4 + 1 Each leading to 16 Substitution of 6 or 8 must be seen 5(b) 2116 2 B1 for 92 92 92 1 2 2 2   + −     or 2162 or 90 90 90 1 1 2 2 2   + + +     or 2070 or 2 90 1 2   +     or 2 91 1 2 +       or 91 91 1 1 2 2 4   + +     or 2115.75 oe Communication: seen in two of the following questions 1 1(b) For showing working, 1 + 1 + ... + 5 2(a) For showing at least three differences 3(a) For showing four differences or rewriting four terms as 1 × 2, 2 × 3, 3 × 4, 4 × 5, etc. or writing out the addition sequences 4(b)(i) For two examples of lengths before different k 4(b)(ii) For two examples of lengths after different k