Cambridge IGCSE Mathematics - International 0607 — 2019 May/June Paper 5 · Variant 2

This document consists of 8 printed pages. DC (RW) 170701/3 © UCLES 2019 [Turn over * 7 5 6 6 2 4 3 4 0 8 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 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. Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education

2 0607/52/M/J/19 © UCLES 2019 Answer all the questions. SQUARE ROOTS WITHIN SQUARE ROOTS This investigation looks at sequences of terms with square roots. You can form a sequence by using square roots within square roots. 6, 6 6 + , 6 6 6 + + , 6 6 6 6 + + + , … You can calculate the first four terms of this sequence as follows. 6 = 2.4494… 6 6 + = . 6 2 4494f + = .8 4494f = 2.9068… 6 6 6 + + = . 6 2 9068f + = .8 9068f = 2.9844… 6 6 6 6 + + + = . 6 2 9844f + = .8 9844f = 2.9974… 1 (a) Complete each part of the calculation below to work out the next term of the sequence, writing each decimal as far as the 4th decimal place. 6 6 6 6 6 + + + + = … 6+ = … = … (b) As the sequence continues, the terms get closer and closer to an integer. This integer is the integer limit of the sequence. Write down the integer limit of this sequence. …

3 0607/52/M/J/19 © UCLES 2019 [Turn over 2 Here is a similar sequence of square roots. 30, 30 30 + , 30 30 30 + + , 30 30 30 30 + + + , … (a) Calculate the first three terms, writing each decimal as far as the 4th decimal place. … , … , … (b) Write down the integer limit of this sequence. …

4 0607/52/M/J/19 © UCLES 2019 3 (a) Complete this table for sequences similar to those in question 1(a) and question 2. 1st term Integer limit 2 6 12 4 20 30 42 7 (b) (i) Use part (a) to find the first term of the sequence that has an integer limit of 8. … (ii) Calculate the 2nd term of the sequence in part (i), writing the decimal as far as the 4th decimal place. …

5 0607/52/M/J/19 © UCLES 2019 [Turn over 4 The general sequence is k, k k + , k k k + + , k k k k + + + , … The integer limit of the sequence is the integer N. For such sequences, ( ) k N N a = - , where a is a constant. (a) Use the last row of the table in question 3(a) to find the value of a. … (b) Use ( ) k N N a = - to show that 90, 90 90 + , 90 90 90 + + , 90 90 90 90 + + + , … has an integer limit of N 10 = . (c) Find the first three terms, in square root form, of the sequence that has an integer limit of N 26 = . … , … , …

6 0607/52/M/J/19 © UCLES 2019 5 Here is the general form of another sequence of square roots with integer limit N. k, k k 2 + , k k k 2 2 + + , k k k k 2 2 2 + + + , … For such sequences, ( ) k N N a = - , where a is a constant. When k 24 = the integer limit of the sequence is N 6 = . (a) Find the value of the constant a. … (b) (i) Find the value of k when the integer limit is N 5 = . … (ii) Write down the 3rd term of the sequence in part (i) in square root form. … (iii) Calculate the 3rd term, writing the decimal as far as the 4th decimal place. …

7 0607/52/M/J/19 © UCLES 2019 [Turn over 6 Here is the general form of another sequence of square roots with integer limit N. k, k k 5 + , k k k 5 5 + + , k k k k 5 5 5 + + + , … For this sequence, ( ) k N N a = - , where a is a constant. The sequence 14, 14 5 14 + , 14 5 14 5 14 + + , 14 5 14 5 14 5 14 + + + , … has an integer limit of N 7 = . Find the value of the constant a. … Question 7 is printed on the next page.

8 0607/52/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. 7 Here is the general sequence with integer limit N. k, k x k + , k x k x k + + , k x k x k x k + + + , … For all such sequences, ( ) k N N a = - , where a is a constant that depends on the value of x. (a) (i) Use question 4, question 5 and question 6 to complete the table. x a Question 4 1 Question 5 Question 6 5 (ii) Write down an expression for k in terms of N and x. … (b) Find the integer limit, N, of this sequence. 7, 7 6 7 + , 7 6 7 6 7 + + , 7 6 7 6 7 6 7 + + + , … …

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 0607/52 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.

0607/52 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.

0607/52 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

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 4 of 5 Question Answer Marks Partial Marks 1(a) 6 2.9974[...] + 1 8.9974[...] 1 FT their copy of 2.9974… 2.9995[…] 1 FT their 8.9974[...] 1(b) 3 1 2(a) 5.4772[…] 5.9562[…] 5.9963[…] 2 B1 for one correct FT their previous term FT their previous term C opportunity 2(b) 6 1 3(a) 1st term Limit 2 2 6 3 12 4 20 5 30 6 42 7 1 3(b)(i) 56 cao 1 C opportunity 3(b)(ii) 7.9676[…] 1 FT their √ if seen in (i) C opportunity 4(a) 1 1 C opportunity 4(b) 10(10 – 1) = 90 oe or 10 × 9 = 90 1 4(c) 650 , 650 650 + , 650 650 650 + + isw 2 B1 for 650 seen C opportunity 5(a) 2 1 C opportunity 5(b)(i) 15 1 C opportunity 5(b)(ii) 15 2 15 2 15 + + isw 1 FT their 15 5(b)(iii) 4.9536[…] 1 FT their correct (b)(ii) using their 15 6 5 1 C opportunity

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 5 of 5 Question Answer Marks Partial Marks 7(a)(i) x a 1 1 2 2 5 5 1 7(a)(ii) N(N – x) oe 1 7(b) 7 1 C opportunity Communication: seen in five of the following questions 2 C1 for three opportunities seen 2(a) 30 30 30 + + or 30 5.4772[...] + or 35.4772[...] 3(b)(i) 8 × 7 [=] 56 or 42 + 14 [=] 56 or 72 + 7 [=] 56 or 82 – 8 [=] 56 or 49 + 9 [=] 56 or 64 – 8 [=] 56 OR Differences seen in left column of table (Qu 3(a) all of 4, 6, 8, 10, 12) OR multiplications 1 × 2, 2 × 3, 3 × 4, 4 × 5, 5 × 6, 6 × 7 or 12 + 1, 22 + 2, 32 + 3, 42 + 4, 52 + 5, 62 + 6 or 22 – 2, 32 – 3, 42 – 4, 52 – 5, 62 – 6, 72 – 7 3(b)(ii) 56 56 + 4(a) 42 = 7 × 6 or 42 = 7(7 – a) and 42 = 49 – 7a or 6 = 7 – a 4(c) 26 × 25 or 26 × (26 – 1) or 262 – 26 5(a) 24 = 6 × 4 or 24 = 6(6 – 2) or 24 = 6(6 – a) and 24 = 36 – 6a or 4 = 6 – a 5(b)(i) 5 × 3 or 5(5 – 2) 6 14 = 7 × 2 or 14 = 7(7 – 5) or 14 = 7(7 – a) and 14 = 49 – 7a or 2 = 7 – a 7(b) Correct calculation of 4th term or a subsequent term or 7 = N (N – 6) oe