Cambridge IGCSE Mathematics - International 0607 — 2020 Oct/Nov Paper 6 · Variant 1

Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 Investigation and Modelling (Extended) October/November 2020 1 hour 40 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer both part A (Questions 1 and 2) and part B (Questions 3 to 6). ● 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 [ ]. DC (LK/CB) 189361/3 © UCLES 2020 [Turn over This document has 12 pages. Blank pages are indicated. * 9 9 6 9 4 0 8 6 1 3 *

2 0607/61/O/N/20 © UCLES 2020 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 and 2) PILING SQUARES (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at different ways of piling squares. All the squares are the same size. 1 Squares are piled in a pattern, like this: 1 square on the bottom row 2 squares on the bottom row 3 squares on the bottom row Total = 1 square Total = 3 squares Total = 6 squares (a) On the dotty paper, complete the next diagram in the sequence. [1]

3 0607/61/O/N/20 © UCLES 2020 [Turn over (b) (i) Complete the table. Number of squares on the bottom row (s) 1 2 3 4 5 6 7 Total number of squares (T) 1 3 6 [2] (ii) A formula for finding the total number of squares is xs ys T 2 = + . Find the value of x and the value of y. x = … y = … [3] (iii) Show that the formula in part (ii) gives the correct total for a pattern of squares with 8 squares on the bottom row. [3]

4 0607/61/O/N/20 © UCLES 2020 2 Black squares and white squares are now piled on top of each other. Every row starts with a white square, followed by a black square, followed by a white square and so on. 1 square on the bottom row 2 squares on the bottom row 3 squares on the bottom row 1 white square 2 white squares 4 white squares 0 black squares 1 black square 2 black squares Total = 1 square Total = 3 squares Total = 6 squares (a) On the dotty paper, complete the next diagram in the sequence. [1] (b) Complete the table. Number of squares on the bottom row (s) 1 2 3 4 5 6 7 Number of black squares (b) 0 1 2 9 Number of white squares (w) 1 2 4 12 Total number of squares (T ) 1 3 6 21 [3]

5 0607/61/O/N/20 © UCLES 2020 [Turn over (c) (i) Complete the two tables for black squares. Some of the information is in the table in part (b). Odd number of squares on the bottom row (s) 1 3 5 7 9 Number of black squares (b) 0 2 20 Even number of squares on the bottom row (s) 2 4 6 8 10 Number of black squares (b) 1 9 16 [1] (ii) A formula for finding the number of black squares, b, when the number of squares on the bottom row, s, is an odd number is b xs y 2 = + . Find the value of x and the value of y. x = … y = … [3] (iii) Find a formula for the number of black squares, b, when the number of squares on the bottom row, s, is an even number. … [2]

6 0607/61/O/N/20 © UCLES 2020 (d) (i) Complete the two tables for white squares. Some of the information is in the table in part (b). Odd number of squares on the bottom row (s) 1 3 5 7 9 Number of white squares (w) 1 4 25 Even number of squares on the bottom row (s) 2 4 6 8 10 Number of white squares (w) 2 12 20 [1] (ii) A formula for finding the number of white squares, w, when the number of squares on the bottom row, s, is an odd number is w xs ys 4 1 2 = + + . Find the value of x and the value of y. x = … y = … [3]

7 0607/61/O/N/20 © UCLES 2020 [Turn over (iii) Find a formula for the number of white squares, w, when the number of squares on the bottom row, s, is an even number. … [3] (e) A pattern has a total of 253 squares. Calculate the number of squares on the bottom row, the total number of white squares and the total number of black squares. Number of squares on the bottom row … Total number of white squares … Total number of black squares … [4]

8 0607/61/O/N/20 © UCLES 2020 B MODELLING (QUESTIONS 3 to 6) A BOUNCING BALL (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at modelling the bounce of a ball. A ball drops vertically onto a hard floor. Each time the ball bounces it rises vertically upwards until its speed becomes zero. At this point the ball drops vertically to bounce on the hard floor again. 3 A ball drops from a height of 10 metres. Each time it bounces it rises to a maximum height which is half the height that it previously dropped. NOT TO SCALE 10 m 5 m 2.5 m 1st bounce 2nd bounce 3rd bounce (a) Calculate the maximum height of the ball after 4 bounces. … [2] (b) (i) Calculate the first maximum height that is less than 10 cm. … [2] (ii) Write down how many bounces the ball has made when its first maximum height is less than 10 cm. … [1]

9 0607/61/O/N/20 © UCLES 2020 [Turn over (c) A model for the maximum height, h metres, which the ball rises after n bounces, is h p qn # = . For example, when n 1 = , h 5 = . Find the value of p and the value of q. p = … q = … [3] 4 A ball drops from a height of 35 metres. After 4 bounces it reaches a maximum height of 0.056 metres. Each time it bounces it rises to a maximum height which is a fraction of the height that it previously dropped. Calculate the value of this fraction. … [3]

10 0607/61/O/N/20 © UCLES 2020 5 A model for the total distance, D metres, which a ball travels vertically, is D p q q 1 1 = - + e o, where p is the height from which the ball is dropped and q is the fraction of the previous maximum height. (a) (i) When q 0 = , find D and explain what happens to the ball. D = … and … … [1] (ii) When q 1 = , explain what happens to the ball. … … [1] (b) Calculate the total distance that the ball in Question 3 travels vertically. … [3] (c) A ball drops from a height of 40 metres. The ball travels a total distance of 200 metres. Each time it bounces it rises to a maximum height which is a fraction of the height that it previously dropped. Calculate the value of this fraction. … [3]

11 0607/61/O/N/20 © UCLES 2020 [Turn over 6 A ball drops from a height of 10 metres. Each time it bounces it rises to a maximum height which is half the height that it previously dropped. Time is measured in seconds. (a) A model for the time taken until the nth bounce, tn, is t q q q 7 10 1 1 2 n n = - + - f p, where q is the fraction of the previous maximum height. (i) Show that t 7 10 1 = . [1] (ii) Show that the time taken until the 10th bounce is 8.0 seconds, correct to 2 significant figures. [2] (b) A model for T, the total time that the ball bounces, is T q q 7 10 1 1 = - + f p. Calculate the total time that the ball bounces. … [2] Questions 6(c) and 6(d) are printed on the next page.

12 0607/61/O/N/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. (c) Sketch the graph of the model for T in part (b), for q 0 1 1 G . 0 0 Time (seconds) Fraction of the previous maximum height T q 1 [4] (d) The time taken until the first bounce, t1, is also modelled by . t p 9 8 2 1 = , where p is the initial height that the ball drops. (i) Find the value of t1 when the ball is dropped from a height of 22.5 m. … [1] (ii) Change the model for the total time T, in part (b), for the ball when it is dropped from a height of 22.5 m. … [1]

This document consists of 7 printed pages. © UCLES 2020 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) October/November 2020 MARK SCHEME Maximum Mark: 60 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 2020 series for most Cambridge IGCSE™, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 2 of 7 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.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 3 of 7 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

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 4 of 7 Question Answer Marks Partial Marks A INVESTIGATION PILING SQUARES 1(a) 1 1(b)(i) s 1 2 3 4 5 6 7 T 1 3 6 10 15 21 28 2 B1 for 3 correct 1(b)(ii) Substitution of two pairs of values from their table or at least 3 second differences of 1 or use of general quadratic with differences or triangle numbers stated C1 FT their 1(b)(i) 1 2 x = oe 1 2 y = oe B2 B1 for each 1(b)(iii) their x × 82 + their y × 8 oe 1 FT their 1(b)(ii) 28 + 8 or difference of 8 seen below table or 1 + 2 + … + 8 or 8 9 2 × or diagram labelled up to 36 B1 36 seen twice nfww B1 2(a) 1 2(b) s 1 2 3 4 5 6 7 b 0 1 2 4 6 9 12 w 1 2 4 6 9 12 16 T 1 3 6 10 15 21 28 3 B2 for 4 or 5 cells correct or B1 for 2 or 3 cells correct

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 5 of 7 Question Answer Marks Partial Marks 2(c)(i) Odd s 1 3 5 7 9 b 0 2 6 12 20 Even s 2 4 6 8 10 b 1 4 9 16 25 1 B1 for 25 2(c)(ii) Substitution of two pairs of values from their table C1 FT their ‘odd’ table in 2(c)(i) 1 4 x = oe 1 4 y = − oe 2 B1 for each 2(c)(iii) b = C1 2 2 s       oe B1 2(d)(i) Odd s 1 3 5 7 9 w 1 4 9 16 25 Even s 2 4 6 8 10 w 2 6 12 20 30 1 B1 for 30 2(d)(ii) Substitution of two pairs of values from their table or trialling (checking) a formula by substitution of two pairs of values from their table or correct equivalent expression or total – black 2 2 1 1 1 1 2 2 4 4 s s s   = + − −     C1 FT their ‘odd’ table in 2(d)(i) FT their 1(b)(ii) − their 2(c)(ii) 1 4 x = 1 2 y = B2 B1 for each

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 6 of 7 Question Answer Marks Partial Marks 2(d)(iii) 2 1 1 4 2 w s s = + oe isw 3 M1 for at least three second differences of 2 seen or use of general quadratic with differences M1 for n2 + n oe or for comparison with ‘odd’ table in 2(c)(i) 2(e) Total formula = 253 soi oe or continuation of sequence C1 FT their x and their y in 1(b)(i) [Bottom row] = 22 [White] = 132 [Black] = 121 3 B1 for each If B0 scored SC1 for 23 B MODELLING A BOUNCING BALL 3(a) 1.25 seen oe or 2.5 4 or 10 16 or 5 8 seen C1 0.625 [m] or 62.5 [cm] B1 3(b)(i) 7.8[1…] or 0.078[1…] or 78.[1…] or 7.8 to 7.813 oe B1 Correct units to match answer C1 3(b)(ii) 7 1 3(c) Substitution of h and n in h = pqn C1 10 1 2 oe B2 B1 for each 4 Correct substitution i.e. 4 0.056 35 q = × C1 1 5 oe nfww B2 M1 for 4 0.056 35 q = or x4 = 35 0.056 or 4 0.056 35 x = 5(a)(i) p and explanation that the ball does not bounce 1 5(a)(ii) The ball bounces up to the same height or the ball never stops bouncing oe 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2020 © UCLES 2020 Page 7 of 7 Question Answer Marks Partial Marks 5(b) 1 1 2 10 1 1 2 their their   +       −   oe C1 FT their value of q in 3(c) 30 B1 FT their 0 < q < 1 from 3(c) m C1 5(c) Correct substitution of 40 and 200 C1 2 3 oe B2 M1 for clearing fraction 6(a)(i) 1 0.5 1 – 0.5 − leading to 10 7 C1 6(a)(ii) 10 10 1 0.5 2 0.5 7 1 0.5   + −     −   C1 8.02… B1 6(b) 10 1 0.5 7 1 0.5   +     −   C1 8.32… or 8.33 B1 6(c) Scale on T axis C1 Correct sketch B3 B1 for general shape curving upwards B1 for intercept within 6 mm of 0 B1 for extension of curve beyond q = 0.8 dependent on first B1 6(d)(i) 15 7 or 2.14 to 2.143 1 6(d)(ii) 1 15 7 1 q T q   + =     −   1 FT their 6(d)(i)