Cambridge IGCSE Mathematics - International 0607 — 2020 May/June Paper 6 · Variant 3

Cambridge IGCSE™ DC (LK/SG) 187926/2 © UCLES 2020 [Turn over This document has 12 pages. Blank pages are indicated. CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 Investigation and Modelling (Extended) May/June 2020 1 hour 40 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer both part A (Questions 1 to 6) and part B (Questions 7 to 11). ● Use a black or dark blue pen. You may use an HB pencil for any diagrams or graphs. ● Write your name, centre number and candidate number in the boxes at the top of the page. ● Write your answer to each question in the space provided. ● Do not use an erasable pen or correction fluid. ● Do not write on any bar codes. ● You should use a graphic display calculator where appropriate. ● You may use tracing paper. ● You must show all necessary working clearly, including sketches, to gain full marks for correct methods. ● In this paper you will be awarded marks for providing full reasons, examples and steps in your working to communicate your mathematics clearly and precisely. INFORMATION ● The total mark for this paper is 60. ● The number of marks for each question or part question is shown in brackets [ ]. * 4 6 8 3 4 5 7 0 7 8 *

2 0607/63/M/J/20 © UCLES 2020 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 6) DIGITAL ROOTS (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation is about the digital roots of positive integers. To find the digital root of a positive integer, add its digits and, if necessary, the digits of the resulting number and so on until a single digit remains. Examples The digital root of 7: = 7 The digital root of 23: 2 + 3 = 5 The digital root of 78: 7 + 8 = 15 1 + 5 = 6 The digital root of 199: 1 + 9 + 9 = 19 1 + 9 = 10 1 + 0 = 1 1 (a) Find the digital root of 2067. … [2] (b) The digital root of 295 is 7. This can be written as D(295) = 7. Find D(173). … [1] (c) Find a 3-digit number with a digital root of 4. … [1]

3 0607/63/M/J/20 © UCLES 2020 [Turn over 2 (a) Write down the maximum value of a digital root. … [1] (b) Find a number, greater than 9500, which will give this maximum value for its digital root. … [2] 3 (a) Use some values of x to find the relationship between D(x) and D(x + 9). … [3] (b) Find the relationship between D(x) and D(x + 9n) where n is a positive integer. [2]

4 0607/63/M/J/20 © UCLES 2020 4 (a) Complete this table. x y D(x) D(y) D(x # y) D(D(x) # D(y)) 63 101 9 2 D(63 # 101) = D(6363) = D(18) = 9 D(9 # 2) = D(18) = 9 315 76 9 4 D(315 # 76) = D(23940) = D(…) = … D(9 # 4) = D(36) = 9 253 42 1 6 D(1 # 6) = D(6) = 6 [3] (b) Write down an algebraic relationship between D(x × y) and D(D(x) × D(y)). … [1] (c) D(x2) = (D(x))2 Is this statement correct? Show how you decide. … because … [2]

5 0607/63/M/J/20 © UCLES 2020 [Turn over 5 The diagram shows some values of D(x3) plotted against values of x from 1 to 10. 0 0 1 2 3 4 5 6 7 8 9 5 10 x D(x3) (a) Complete the diagram. [2] (b) Find the nth term of the sequence of values of x for which D(x3) = 8 . … [2] (c) Use digital roots to decide whether 1 000 030 300 106 031 030 301 is a cube number. Give a reason for your answer. [2]

6 0607/63/M/J/20 © UCLES 2020 6 (a) Pn is the nth prime number. The diagram shows the value of D(Pn) for the 5th to the 50th prime number. 0 0 5 10 15 20 25 30 35 40 45 50 1 2 3 4 5 6 7 8 9 D(Pn) n (i) Complete the diagram for the first four prime numbers. [1] (ii) Is it possible to use the diagram to predict D(P51)? Give a reason for your answer. … [1]

7 0607/63/M/J/20 © UCLES 2020 [Turn over (b) This diagram shows the frequency of the digital roots for the first 1200 prime numbers. 1 2 3 4 5 Digital root Frequency 0 50 100 150 200 6 7 8 9 (i) Write down two observations from the diagram about the digital roots of these prime numbers. 1 … 2 … [2] (ii) 4 … 27 is a 4-digit number which is not a prime number. Use the diagram to find a possible missing digit. … [2]

8 0607/63/M/J/20 © UCLES 2020 B MODELLING (QUESTIONS 7 TO 11) EARTHQUAKES (30 marks) You are advised to spend no more than 50 minutes on this part. The task is about the strength and frequency of earthquakes and the probability of their occurrence. The strength of an earthquake is measured in magnitudes. An increase in magnitude of 1 increases the energy released by the earthquake by a factor of 32. Example A magnitude 4.7 earthquake releases 32 times as much energy as a magnitude 3.7 earthquake. 7 (a) Write down the magnitude of an earthquake that releases 32 times the energy of a magnitude 2.5 earthquake. … [1] (b) A magnitude 6 earthquake releases 30 000 units of energy. Calculate the number of units of energy a magnitude 7 earthquake releases. … [2]

9 0607/63/M/J/20 © UCLES 2020 [Turn over 8 A model for the energy, E, that an earthquake releases is E = g # h1.5M where g and h are constants and M is the magnitude of the earthquake. (a) An earthquake of magnitude 6 releases 30 000 units of energy. Write an equation involving g and h. … [1] (b) An earthquake of magnitude 8 releases 30 000 000 units of energy, correct to 1 significant figure. Write an equation involving g and h. … [1] (c) Use part (a) and part (b) to find (i) the value of h, … [3] (ii) the value of g. … [2] (d) The magnitude of an earthquake is 6.2 . Calculate the number of units of energy that it releases. … [2]

10 0607/63/M/J/20 © UCLES 2020 9 This table shows information about the number of earthquakes in northern Chile between April 2008 and April 2018. Minimum magnitude (M) Number of earthquakes (N) 3.5 2028 4.0 1912 4.5 784 5.0 230 5.5 57 6.0 14 6.5 3 7.0 0 There were a total of 2028 earthquakes with . M 3 5 H . There were a total of 2028 - 1912 = 116 earthquakes with a magnitude in the range . . M 3 5 4 0 1 G . (a) Find the number of earthquakes in the range . M 5 6 5 1 G . … [2] (b) A model for this data is N M k = , where k is a constant and N is the number of earthquakes with minimum magnitude M. Is this a suitable model? Show how you decide. [2]

11 0607/63/M/J/20 © UCLES 2020 [Turn over 10 Another model for these earthquakes is . log cM N 7 15 = + where c is a constant. (a) Complete the table for logN, correct to 1 decimal place. M N logN 3.5 2028 3.3 4.0 1912 3.3 4.5 784 2.9 5.0 230 2.4 5.5 57 6.0 14 6.5 3 [2] (b) Complete this scatter diagram of logN against M. The first four points have been plotted for you. 4 3.5 3 2.5 2 1.5 1 0.5 0 3 3.5 4 4.5 5 5.5 6 6.5 7 logN M [1] (c) (i) The mean point is (5, 2.2). On the diagram, draw a line of best fit. [1] (ii) Use your line of best fit to find the value of c. … [2] Question 11 is printed on the next page.

12 0607/63/M/J/20 © UCLES 2020 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. 11 A model for the number of earthquakes, N, in San Francisco between 1950 and 2018 is N = 10 (6.6-0.91M), where M is the minimum magnitude. (a) There were 1013 earthquakes with a minimum magnitude of 4 during this time. Find the difference between this actual number and the number that the model predicts. … [2] (b) Use the model to estimate the total number of earthquakes of any magnitude. … [2] (c) (i) On the diagram, sketch the graph of N for . . M 3 5 7 0 G G . 0 3.5 7 N M [3] (ii) What effect would another earthquake of magnitude 7.0 in this period have on the graph? … [1]

This document consists of 6 printed pages. © UCLES 2020 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2020 MARK SCHEME Maximum Mark: 60 Published Students did not sit exam papers in the June 2020 series due to the Covid-19 global pandemic. This mark scheme is published to support teachers and students and should be read together with the question paper. It shows the requirements of the exam. The answer column of the mark scheme shows the proposed basis on which Examiners would award marks for this exam. Where appropriate, this column also provides the most likely acceptable alternative responses expected from students. Examiners usually review the mark scheme after they have seen student responses and update the mark scheme if appropriate. In the June series, Examiners were unable to consider the acceptability of alternative responses, as there were no student responses to consider. Mark schemes should usually be read together with the Principal Examiner Report for Teachers. However, because students did not sit exam papers, there is no Principal Examiner Report for Teachers for the June 2020 series. Cambridge International will not enter into discussions about these mark schemes. Cambridge International is publishing the mark schemes for the June 2020 series for most Cambridge IGCSE™ and Cambridge International A & AS Level components, and some Cambridge O Level components.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 2 of 6 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.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 3 of 6 Maths-Specific Marking Principles 1 Unless a particular method has been specified in the question, full marks may be awarded for any correct method. However, if a calculation is required then no marks will be awarded for a scale drawing. 2 Unless specified in the question, answers may be given as fractions, decimals or in standard form. Ignore superfluous zeros, provided that the degree of accuracy is not affected. 3 Allow alternative conventions for notation if used consistently throughout the paper, e.g. commas being used as decimal points. 4 Unless otherwise indicated, marks once gained cannot subsequently be lost, e.g. wrong working following a correct form of answer is ignored (isw). 5 Where a candidate has misread a number in the question and used that value consistently throughout, provided that number does not alter the difficulty or the method required, award all marks earned and deduct just 1 mark for the misread. 6 Recovery within working is allowed, e.g. a notation error in the working where the following line of working makes the candidate’s intent clear. 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

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 4 of 6 Question Answer Marks Partial Marks A INVESTIGATION DIGITAL ROOTS 1(a) 2 + 6 + 7 C1 6 B1 1(b) 2 1 1(c) Any 3-digit number with a digital root of 4 1 2(a) 9 1 2(b) Any number > 9500 with a digital root of their 9 2 FT their 9 3(a) At least two correct pairs of examples C2 C1 for one correct example D(x + 9) = D(x) oe B1 3(b) D(x) = D(x + 9n) B1 At least one example with n > 1 C1 4(a) D(315 × 76) = D(23 940) = D(18) = 9 D(253 × 42) = D(10 626) = D(15) = 6 B2 B1 for each correct digital root nfww Two lines of correct working in the bottom row C1 4(b) D(x × y) = D(D(x) × D(y)) 1 4(c) No, any correct example 2 C1 for two correct trials 5(a) Points correctly plotted 2 B1 for three or four points correctly plotted 5(b) 3n – 1 oe 2 B1 for 3n + k or jn – 1 (j≠0) 5(c) No and finding the digital root is 7 and referring to the diagram 2 B1 for digital root is 7.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 5 of 6 Question Answer Marks Partial Marks 6(a)(i) Points correctly plotted. 1 6(a)(ii) No, patterns not repeated oe. 1 6(b)(i) Any two of • 1, 2, 4, 5, 7, 8 occur with about the same frequency oe • 6 and 9 do not occur at all oe • 3 occurs very infrequently oe 2 C1 for listing any one 6(b)(ii) 2 or 5 B1 Indicates digital root of 6 or 9 or finds digital roots with 2, 4, 7 and another digit C1 B MODELLING EARTHQUAKES 7(a) 3.5 1 7(b) 960 000 2 B1 for figs 96 8(a) 30 000 = g × h9 1 8(b) 30 000 000 = g × h12 1 8(c)(i) Sight of relevant division C1 [h =] 10 A2 M1 for 1000 = h3 8(c)(ii) Correct substitution of their h C1 [g =] 3 × 10–5 oe B1 8(d) Correct substitution seen C1 60 000 or 59 900 or 59 860 or 59 857[. …] B1 9(a) 230 – k or subtracting 3 C1 227 B1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2020 © UCLES 2020 Page 6 of 6 Question Answer Marks Partial Marks 9(b) No, with correct working 2 C1 for correct product of two or more M and N (oe division) 10(a) 1.8 1.1 0.5 2 B1 for two correct but not necessarily to 1 dp. 10(b) Correct diagram 1 10(c)(i) Ruled line of best fit passing through (5, 2.2) B1 10(c)(ii) For correct method to find gradient or substitution of point on their line C1 [c =] –1.1 to –0.8 B1 11(a) 10(6.6 – 0.91×4) C1 101 A1 11(b) 3 980 000 or 3 981 071 to 3 981 072 2 M1 for M = 0 soi 11(c)(i) B2 B1 for correct shape (concave – gradient decreasing) Scale on N axis implying intercept at approx. 2600 C1 11(c)(ii) None oe 1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 log(N) Earthquake magnitude M 0 3000 3.5 7 N M