Cambridge IGCSE Mathematics - International 0607 — 2019 Oct/Nov Paper 5 · Variant 1

This document consists of 7 printed pages and 1 blank page. DC (LK) 170791/3 © UCLES 2019 [Turn over * 0 3 2 6 6 5 9 5 5 2 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 Paper 5 (Core) October/November 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 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 Assessment International Education Cambridge International General Certificate of Secondary Education

2 0607/51/O/N/19 © UCLES 2019 Answer all the questions. INVESTIGATION DECIMAL FORMS This investigation looks at the patterns when changing a fraction to its decimal form. Examples . . 3 2 0 666 0 6 f = = o This is a repeating decimal. . 4 3 0 75 = This is a terminating decimal. The fraction 8 5 has a numerator of 5 and a denominator of 8. 1 This question is about terminating decimals. (a) (i) Complete these equivalent fractions. 2 1 10 5 = 5 1 10 2 = 20 7 100 = 25 1 100 = 500 3 1000 = (ii) The denominators of the equivalent fractions in part (i) are 10, 100 and 1000. The smallest prime number is 2. Put a prime number in each box to complete these statements. 10 = = 2 # 5 100 = 10 # 10 = 2 # 5 # # 1000 = 10 # 10 # 10 = 2 # 5 # # # # (iii) Complete the table. Fraction 2 1 5 1 20 7 25 1 500 3 Decimal 0.5 0.2 (iv) Write down a different fraction with a numerator of 1 and a denominator between 30 and 99 which can be written as a terminating decimal. …

3 0607/51/O/N/19 © UCLES 2019 [Turn over (b) (i) Put a prime number in each box to complete these statements. 20 = 2 # 2 # 5 25 = 5 # 5 50 = 2 # # 5 100 = # # 5 # 5 500 = 2 # 2 # # # (ii) Use your answers to part (i) to help you complete the table. Fraction Decimal Number of decimal places Denominator written as a product of primes using powers Larger power 20 1 0.05 2 2 5 2 # 2 25 7 0.28 2 52 2 0 5 9 0.18 2 0 19 10 0.19 2 200 13 0.065 3 2 5 3 2 # 3 0 1 50 1 0.022 17 5000 0.0034 2 5 3 4 # 4 (iii) A fraction has a numerator of 1 and a denominator of 2 5 14 7 # . Write down the number of decimal places in the decimal form of this fraction. … (iv) The denominator of a fraction that can be written as a terminating decimal only has one or two possible prime factors. Write down these prime factors. … and …

4 0607/51/O/N/19 © UCLES 2019 2 This question is about repeating decimals. The number of digits in the repeating pattern is called the repeat length. Example . . 13 1 0 076923 076923076923 0 076 923 f = = o o A C BBBB This is a repeating decimal with a repeat length of 6. (a) (i) Complete these equivalent fractions. 3 1 9 = 11 1 99 = 37 1 999 = 111 1 999 = 41 1 99999 = 7 1 999999 = (ii) Complete the table. Fraction 1 3 11 1 37 1 1 111 41 1 7 1 Decimal .0 3o .0 09 o o .0 027 o o .0 142857 o o Repeat length 1 2 3 5 6 (iii) Use your answers to part (i) and part (ii) to help you complete the table. Fraction Decimal Repeat length Denominator of equivalent fraction 1 3 .0 3o 1 9 = 10 1 1 - 11 1 .0 09 o o 2 99 = 10 1 2 - 37 1 .0 027 o o 999 = 1 111 3 999 = 41 1 5 99 999 = 7 1 .142857 0 o o 6 999 999 = (iv) Give an example of a fraction with a numerator of 1 which can be written as a repeating decimal with a repeat length of 9. …

5 0607/51/O/N/19 © UCLES 2019 [Turn over (v) A repeating decimal has a repeat length of k. Write down an expression, in terms of k, for the denominator of this fraction. … (b) (i) 407 1 11 37 1 11 1 37 1 # # = = 407 1 is changed to its decimal form. Show that this has a repeat length that is equal to the lowest common multiple (LCM) of the repeat lengths of the decimal forms of 11 1 and 37 1 . (ii) Show how the lowest common multiple (LCM) of the repeat lengths of 7 1 and 37 1 gives the repeat length of 259 1 .

6 0607/51/O/N/19 © UCLES 2019 3 Some decimals have non-repeating decimal parts followed by repeating decimal parts. Example . . 0 65 0 65555f = o In this decimal, the 6 does not repeat but the 5 does. (a) Show that adding the decimal forms of 5 1 and 1 3 gives a decimal of this type. (b) Complete the table. Fraction Decimal Number of non- repeating decimal places Repeat length Denominator written as a product of primes using powers 6 1 .0 16o 1 1 2 3 # 12 1 .0 083o 2 1 75 7 24 11 3 600 317 .0 528 3o 2 5 3 3 2 # # 1320 1 .0 000 75 o o 3 2 2 5 3 # 11 3 # # 101750 50001 .0 491410 319 o o 3 6 2 5 11 37 3 # # #

7 0607/51/O/N/19 © UCLES 2019 (c) A fraction of the form c d 2 5 1 a b # # # where a and b are positive integers and c and d are different primes is changed to its decimal form. Using your answers to question 1(b) and question 2(b), explain how to find the number of non-repeating decimal places and the repeat length. … … … …

8 0607/51/O/N/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 0607/51 Paper 5 (Core) October/November 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 October/November 2019 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 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.

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 4 of 5 Question Answer Marks Partial Marks 1(a)(i) 35 4 6 1 C opportunity 1(a)(ii) 2 × 5 2 × 5 × 2 × 5 2 × 5 × 2 × 5 × 2 × 5 1 1(a)(iii) 0.35 0.04 0.006 1 1(a)(iv) Any one of 1 1 1 1 1 , , , , 32 40 50 64 80 1 1(b)(i) 5 2 2 5 5 5 1 1(b)(ii) 2 22 × 5 2 2 52 2 2 2[1] × 52 2 2 22 × 52 2 3 23 × 52 3 3 22 × 53 3 4 23 × 54 4 2 FT their (b)(i) Strict FT their column 4 to get column 5. B1 for 5, 6 or 7 correct entries 1(b)(iii) 14 1 1(b)(iv) 2 and 5 1 2(a)(i) 3, 9, 27, 9, 2439, 142857 1 C opportunity 2(a)(ii) 0.009   0.02439   3 1 2(a)(iii) 0.027   3 999 =103 – 1 0.009   3 999 =103 – 1 0.02439   5 99 999 =105 − 1 0.142857   6 999 999 =106 – 1 1

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 5 of 5 Question Answer Marks Partial Marks 2(a)(iv) e.g. 1 999999999 or 1 111111111 or 1 333 333 333 or 1 81 … 1 2(a)(v) 10k – 1 1 2(b)(i) 0.002457   oe and LCM (2, 3) = 6 oe 1 2(b)(ii) 259 7 37 = × oe and LCM (3, 6) = 6 oe and 0.003861 i i oe 2 B1 for any two of these statements 3(a) 0.2 0.3 0.53 + = i i 1 3(b) 22 × 3 0.093 2 1 52 × 3 0.4583 1 23 × 3 3 1 3 B2 for 8 or 9 correct entries or B1 for 6 or 7 correct entries C opportunity 3(c) The number of non-repeating decimal places is given by the higher of a and b oe and The repeat length is given by the LCM of the repeat lengths of 1 c and 1 d oe 2 B1 for each Communication: Seen in one of the following questions 1 1(a)(i) for method to find equivalent fractions 2(a)(i) for 999 ÷ 37 or 99 999 ÷ 41 or 999 999 ÷ 7 seen or method to find equivalent fractions 3(b) for method for finding factors, e.g. factor trees