Cambridge IGCSE Mathematics - International 0607 — 2018 May/June Paper 5 · Variant 1

* 9 5 9 4 9 5 0 3 2 5 * This document consists of 7 printed pages and 1 blank page. DC (KS) 153522/2 © UCLES 2018 [Turn over CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) May/June 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 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/M/J/18 © UCLES 2018 Answer all the questions. INVESTIGATION LARGEST PRODUCTS This investigation looks at finding the largest product when two or more positive integers have a given sum. For the positive integers 2 and 5 • the sum 2 + 5 is 7 • the product 2 × 5 is 10. 1 (a) Complete this table for all the different pairs of positive integers that have a sum of 8. Integers Sum Product 1 8 2 8 3 8 4 4 8 16 Write down the calculation that gives the largest product. … (b) Complete this table for all the different pairs of positive integers that have a sum of 10. Note that 3 and 7 is the same pair as 7 and 3. Integers Sum Product 10 10 3 7 10 21 10 10 Write down the calculation that gives the largest product. …

3 0607/51/M/J/18 © UCLES 2018 [Turn over (c) Find the largest product of two positive integers that have a sum of 6. … (d) Use your answers to part (a), part (b) and part (c) to help you complete the table. Sum 6 8 10 12 14 Largest product 49 (e) (i) The sum of two positive integers is S. S is an even number. Find an expression, in terms of S, for the largest product of the two integers. … (ii) The sum of two positive integers is 62. Find the largest product of the two integers. … (f) The sum of two positive integers is S. S is an even number. The largest product of the two integers is 576. Find the value of S. …

4 0607/51/M/J/18 © UCLES 2018 2 (a) Complete this table for all the different pairs of positive integers that have a sum of 9. Note that 2 and 7 is the same pair as 7 and 2. Integers Sum Product 9 9 9 9 Write down the calculation that gives the largest product. … (b) Find the largest product of two positive integers that have a sum of 7. …

5 0607/51/M/J/18 © UCLES 2018 [Turn over (c) Use your answers to part (a) and part (b) to help you complete the table. Sum 7 9 11 13 101 Largest product 30 (d) The sum of two positive integers is S. S is an odd number. (i) Explain why the largest product of the two integers is always even. … … (ii) Find an expression, in terms of S, for the largest product of the two integers. Do not simplify your answer. …

6 0607/51/M/J/18 © UCLES 2018 3 (a) Three positive integers have a sum of 6. Complete the table for all the different sets of positive integers that have a sum of 6. Writing the positive integers in a different order does not give a different set. Integers Sum Product 6 6 6 Write down the calculation that gives the largest product. … (b) Look at how you found the largest product in part (a) and in question 1(a). Four positive integers have a sum of 40. Show that the largest product of these four integers is 10 000.

7 0607/51/M/J/18 © UCLES 2018 (c) Complete the table. Sum 6 40 15 24 Number of positive integers in the sum 3 4 5 6 Largest product 10 000 (d) n integers have a sum of 40, where n is a factor of 40. Find the value of the largest product of the n integers. …

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IGCSE™ is a registered trademark. 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 0607/51 Paper 5 (Core) May/June 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 May/June 2018 series for most Cambridge IGCSE™, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 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.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 4 of 5 Question Answer Marks Partial Marks 1(a) 1 7 8 7 2 6 8 12 3 5 8 15 4 4 8 16 4 × 4 2 B1 for one correct row. 1(b) 1 9 10 9 2 8 10 16 3 7 10 21 4 6 10 24 5 5 10 25 5 × 5 2 B1 for two correct rows. 1(c) 9 1 C opportunity 1(d) 6 8 10 12 14 their 9 16 25 36 49 1 1(e)(i) 2 2       S oe isw 1 1(e)(ii) 961 1 FT their formula of the form 2 S k k > 1 C opportunity 1(f) 48 1 C opportunity 2(a) 1 8 9 8 2 7 9 14 3 6 9 18 4 5 9 20 4 × 5 or 5 × 4 2 B1 for two correct rows 2(b) 12 1 C opportunity 2(c) 7 9 11 13 101 their 12 20 30 42 2550 2 B1 for 20 and 42 B1 for 2550 C opportunity 2(d)(i) even × odd = even oe 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 5 of 5 Question Answer Marks Partial Marks 2(d)(ii) 1 1 2 2 2 2    − +       S S oe isw 1 3(a) 1 1 4 6 4 1 2 3 6 6 2 2 2 6 8 2 × 2 × 2 2 B1 for one correct row 3(b) 40 ÷ 4 = 10 oe B1 10 × 10 × 10 × 10 or 104 oe B1 3(c) 6 40 15 24 3 4 5 6 8 10 000 243 4096 2 B1 for two correct cells C opportunity 3(d) 1 048 576 1 C opportunity Communication: Seen in four of the following questions 1 1(c) 3 × 3 seen 1(e)(ii) 31 seen in a relevant calculation 1(f) 24 seen or trial and improvement. 2(b) 3 × 4 or 4 × 3 seen 2(c) 6 × 7 or 7 × 6 or 50 × 51 or 51 × 50 seen 3(c) 35 or 3 × 3 × 3 × 3 × 3 or 46 or 4 × 4 × 4 × 4 × 4 × 4 3(d) 202 or 58 or 85 or 40       n n