Cambridge IGCSE Mathematics - International 0607 — 2019 May/June Paper 5 · Variant 3

This document consists of 10 printed pages and 2 blank pages. DC (KS/CGW) 170702/2 © UCLES 2019 [Turn over * 6 2 3 4 2 8 2 3 5 8 * Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 2019 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.

2 0607/53/M/J/19 © UCLES 2019 Answer all the questions. INVESTIGATION JUMPING FROGS This investigation looks at the number of different ways that a frog can jump between stones in a line. The stones are always 1 unit apart. The frog always jumps • from left to right • from the first stone to the last stone. The diagram shows a frog sitting on a stone in a pond. First stone Last stone This frog has a jump length of 1 unit. This is enough to move from one stone to the next stone. There is only 1 way to jump between two stones. First stone Last stone There is only 1 way to jump between three stones. When there are more than three stones in a line, there is always only 1 way for this frog to jump from the first stone to the last stone.

3 0607/53/M/J/19 © UCLES 2019 [Turn over 1 A different frog has a maximum jump length of 2 units. There is still only 1 way to jump between two stones. There are now 2 ways to jump between three stones. (a) Complete the diagrams below to show the 3 ways that this frog can jump between four stones.

4 0607/53/M/J/19 © UCLES 2019 (b) Complete the diagrams below to show the 5 ways that this frog can jump between five stones.

5 0607/53/M/J/19 © UCLES 2019 [Turn over (c) The table shows the number of ways to jump between stones when the maximum jump length is 2 units. Number of stones Number of ways 2 1 3 2 4 3 5 5 6 8 7 13 The numbers in the last column of the table form a sequence. (i) Write down the addition rule to find further terms in this sequence. … … (ii) Use your rule to show that the number of ways for this frog to jump between 8 stones is 21.

6 0607/53/M/J/19 © UCLES 2019 2 Another frog has a maximum jump length of 3 units. There is only 1 way for this frog to jump between two stones. There are 2 ways to jump between three stones. (a) There are 4 ways to jump between four stones. These are the same 3 ways as in question 1(a) and 1 new way. Draw the new way on the diagram below. (b) There are 7 ways to jump between five stones. These are the same 5 ways as in question 1(b) and 2 new ways. Draw the 2 new ways on the diagrams below.

7 0607/53/M/J/19 © UCLES 2019 [Turn over (c) The table shows the number of ways to jump between stones when the maximum jump length is 3 units. Number of stones Number of ways 2 1 3 2 4 4 5 7 6 13 7 24 The numbers in the last column of the table form a sequence. (i) Write down the rule to find further terms in this sequence. … … (ii) Use your rule to find the number of ways for this frog to jump between 8 stones.

8 0607/53/M/J/19 © UCLES 2019 3 The table shows the number of ways to jump between 2 to 8 stones when the maximum jump length is 1 to 6 units. Number of stones Maximum jump length 1 unit 2 units 3 units 4 units 5 units 6 units 2 1 1 1 1 1 1 3 1 2 2 2 2 2 4 1 3 4 4 4 4 5 1 5 7 8 8 8 6 1 8 13 15 16 16 7 1 13 24 29 31 32 8 1 21 (a) Complete the table. (b) The numbers in the column for maximum jump length of 4 units form a sequence. Write down the rule for finding further terms in this sequence. … … (c) The numbers in the column for the maximum jump length of 5 units also form a sequence. Write down the rule for finding further terms in this sequence. … …

9 0607/53/M/J/19 © UCLES 2019 [Turn over 4 This is the same table as in question 3, with some numbers missing. Some of the numbers have been written as powers of 2. Number of stones Maximum jump length 1 unit 2 units 3 units 4 units 5 units 6 units 2 20 3 1 21 21 21 21 21 4 1 3 5 1 5 7 6 1 8 13 15 7 1 13 24 29 31 8 1 21 (a) Copy your values from the table in question 3 into the row for 8 stones. Complete the table by writing in the remaining values as powers of 2. (b) Use patterns from your table to write down, as a power of 2, the number of ways of jumping between 30 stones when the maximum jump length is 29 units. … (c) Find the number of ways of jumping between 30 stones when the maximum jump length is 28 units. …

10 0607/53/M/J/19 © UCLES 2019 (d) From the table, the largest number of ways of jumping between 3 stones is 21. Write down, as a power of 2, the largest number of ways of jumping between (i) 5 stones, … (ii) 20 stones, … (iii) x stones. … (e) (i) Find the smallest jump length that gives your answer in part (d)(i). … (ii) Find an expression in terms of x for the smallest jump length that gives your answer in part (d)(iii). …

11 0607/53/M/J/19 © UCLES 2019 BLANK PAGE

12 0607/53/M/J/19 © UCLES 2019 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 Assessment International Education Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cambridgeinternational.org after the live examination series. Cambridge Assessment International Education is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge. BLANK PAGE

This document consists of 5 printed pages. © UCLES 2019 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 2019 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 2019 series for most Cambridge IGCSE™, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 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.

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 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

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 4 of 5 Question Answer Marks Partial Marks 1(a) 1 1(b) 2 B1 for any two correct 1(c)(i) add the previous two terms oe or add last 2 differences to current number of ways oe or × by 2 subtract number of ways two removed oe 1 1(c)(ii) 8 + 13 = 21 1 2(a) 1 2(b) 1 2(c)(i) add the previous three terms oe or × by 2 then subtract the number of ways that is 3 before oe or add last 3 differences to new number of ways oe 1

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 5 of 5 Question Answer Marks Partial Marks 2(c)(ii) 44 1 C opportunity 3(a) 8 1 21 44 56 61 63 2 B1 for any two of 56, 61, 63 in correct positions C opportunity 3(b) Sum of the previous four terms oe or × by 2 and subtract the number of ways that is 4 before oe 1 3(c) Sum of the previous five terms oe or × by 2 and subtract the number of ways that is 5 before oe 1 4(a) Number of stones Maximum jump length 1 unit 2 units 3 units 4 units 5 units 6 units 2 20 20 20 20 20 20 3 1 21 21 21 21 21 4 1 3 22 22 22 22 5 1 5 7 23 23 23 6 1 8 13 15 24 24 7 1 13 24 29 31 25 8 1 21 44 56 61 63 3 B1 for 20 row complete B1 for any other two rows with powers of 2 complete 4(b) 228 1 4(c) 228–1 or 268 435 455 1 FT their(b) 4(d)(i) 23 1 4(d)(ii) 218 1 4(d)(iii) 2x–2 oe 1 4(e)(i) 4 1 4(e)(ii) x – 1 1 Communication: seen in one of the following questions. 1 2(c)(ii) 7 + 13 + 24 3(a) for showing one summation