Cambridge IGCSE Mathematics - International 0607 — 2019 Oct/Nov Paper 6 · Variant 2

* 0 2 5 8 9 0 8 0 5 8 * This document consists of 15 printed pages and 1 blank page. DC (NH/SW) 170793/2 © UCLES 2019 [Turn over CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) October/November 2019 1 hour 30 minutes 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 both parts A (Questions 1 to 6) and B (Questions 7 to 11). 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 40. Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education

2 0607/62/O/N/19 © UCLES 2019 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 6) CROSSING POINTS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation is about how many times lines cross when you join two rows of dots. Dots are arranged along two parallel lines, A and B. Straight lines join each dot on A to each dot on B. A crossing point is where exactly two straight lines cross. Example When there are 2 dots on A and 4 dots on B there are 6 crossing points. A 2 dots B 4 dots You must complete all diagrams accurately using a ruler and a sharp pencil. 1 When there is 1 dot on A and 1 dot on B, there is no crossing point. A B (a) Each of these diagrams has 1 dot on A. Complete the diagrams by joining the dot on A to each dot on B. A B A B (b) The number of dots on B is b. The number of crossing points is p. Complete this statement. When there is 1 dot on A and b dots on B, then the value of p is … .

3 0607/62/O/N/19 © UCLES 2019 [Turn over 2 There are now 2 dots on A. (a) Complete the diagrams for 1 dot on B and 3 dots on B, and use your results to complete the table. A B A B A B A B A B Number of dots on B (b) Number of crossing points (p) 1 2 1 3 4 6 5 10 (b) Find a formula for p in terms of b when there are 2 dots on A. …

4 0607/62/O/N/19 © UCLES 2019 3 There are now 3 dots on A. (a) Complete each diagram and use your results to complete the table. The first diagram has been completed for you. A B A B A B A B Number of dots on B (b) Number of crossing points (p) 1 2 3 9 4 5 30 (b) Find a formula for p in terms of b when there are 3 dots on A. …

5 0607/62/O/N/19 © UCLES 2019 [Turn over 4 There are now 4 dots on A. This table shows the number of crossing points. Number of dots on B (b) Number of crossing points (p) 1 0 2 6 3 18 4 36 5 60 Find a formula for p in terms of b when there are 4 dots on A. …

6 0607/62/O/N/19 © UCLES 2019 5 Use your answers to question 2(b), question 3(b) and question 4 to help you complete the table when there are b dots on B. Number of dots on A Number of crossing points (p) Factorised form for number of crossing points (p) 1 0 0 2 b b 2 2 2 - ( ) b b 2 - 3 b b 2 2 2 - ( ) b b 2 - 4 b b 2 2 2 - ( ) b b 2 - 5 b b 2 10 2 10 2 - ( ) b b 2 10 1 - a b b 2 2 2 - ( ) ( ) a a b b 2 2 # - -

7 0607/62/O/N/19 © UCLES 2019 [Turn over 6 (a) There are now a dots on A. When the number of dots on B is the same as the number of dots on A, show that your factorised form from question 5 becomes ( ) a a 4 1 2 2 - . (b) (i) Show that the expression in part (a) does not give the correct number of crossing points for the dots in this diagram. A B (ii) Give a reason why this happens. … …

8 0607/62/O/N/19 © UCLES 2019 (c) Equal numbers of dots are marked on A and B so that the expression in part (a) will give the correct number of crossing points. (i) Explain why there cannot be 50 crossing points. … … (ii) There are 29 241 crossing points. Find the number of dots on A. …

9 0607/62/O/N/19 © UCLES 2019 [Turn over The modelling task starts on the next page.

10 0607/62/O/N/19 © UCLES 2019 B MODELLING (QUESTIONS 7 to 11) VALUE OF A CAR (20 marks) You are advised to spend no more than 45 minutes on this part. This task is about how the value of a car changes with time. The value of any car changes continuously. 7 Anna says that this is a model for the value of a car as it gets older. At the end of each year, the value of the car will be half of its value at the start of the year. Eddie pays $16 000 for a new car. (a) Use Anna’s model to complete the table. Age of car (x years) 0 1 2 3 4 5 6 Value (V dollars) 16 000 8000

11 0607/62/O/N/19 © UCLES 2019 [Turn over (b) Use these values to draw a graph of V against x. 0 0 2000 4000 6000 8000 10 000 12 000 14 000 16 000 1 2 3 Age of car (years) Value of car (dollars) 4 5 6 V x (c) Complete this model for V in terms of x. V = 16 000 × … (d) Using Anna’s model, find the value of Eddie’s car when it is 10 years old. …

12 0607/62/O/N/19 © UCLES 2019 8 John’s Cars model the value of a car in this way. Each year, the value of the car decreases by a percentage of its value when it was new. The table shows the value of the car as a percentage of its value when it is new. Age of car (x years) 0 1 2 3 4 5 6 Percentage of value when new (P) 100 72 58 48 44 42 40 (a) On the grid, plot the values of P against x. 0 0 10 20 30 40 50 60 70 80 90 100 1 2 3 Age of car (years) Percentage 4 5 6 P x (b) Show which of the two models gives a higher value for Eddie’s car when it is 5 years old.

13 0607/62/O/N/19 © UCLES 2019 [Turn over 9 Eddie models the percentage values used by John’s Cars. . . log P x x 56 2 15 6 0 for 10 2 = - . (a) On the axes, sketch the graph of this model. Age of car (years) 100 0 0 10 Percentage P x (b) How well does this model fit the percentages used by John’s Cars for cars that are less than 3 years old? …

14 0607/62/O/N/19 © UCLES 2019 10 Lia makes a different model ( ) P ax 1 100 = + . (a) The table in question 8 shows that P = 72 when x = 1. These values fit Lia’s model. Find the value of a, correct to 1 decimal place. … (b) On the axes in question 9, sketch the graph of this model. (c) How well does Lia’s model fit the percentages used by John’s Cars? … …

15 0607/62/O/N/19 © UCLES 2019 11 (a) Use Lia’s model to write a model for the value, $V, of a car that costs $C when it is new. … (b) Use this model to find the age of Eddie’s car when its value is $6500. Give your answer in years and months, correct to the nearest month. …

16 0607/62/O/N/19 © UCLES 2019 BLANK PAGE 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.

This document consists of 8 printed pages. © UCLES 2019 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) October/November 2019 MARK SCHEME Maximum Mark: 40 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.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 2 of 8 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.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 3 of 8 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

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 4 of 8 Question Answer Marks Partial Marks A INVESTIGATION CROSSING POINTS 1(a) Both diagrams correct 1 1(b) 0 1 2(a) Both diagrams correct 0 1 3 6 10 1 2(b) 2 1 1 oe 2 2 = − p b b 2 B1 for 2 2 1 b – 1 2 b oe 3(a) All diagrams correct 0 3 9 18 30 2 B1 for 0 and 3 B1 for 18 3(b) 2 3 3 2 2 = − b b p oe 2 B1 for 3 2 or 1.5 as a coefficient 4 2 6 6 2 2 = − b b p oe 1 FT their 2(b) × 6 5 Number of crossing points (p) Factorised form for number of crossing points (p) 0 0 ….b2 –…. b ( ) 1 2 b b − ….b2 – ….b ( ) 3 1 2 b b − 2 2 2 b b − 6 6 ( ) 2 b b − 6 1 2 10 10 2 2 b b − ( ) 10 1 2 b b − 2 2 2 b b − 2 2 1 1 1 1 2 2 2 2 a a a a − − ( ) ( ) 2 2 a a b b − − × 1 1 3 B1 for row 4 B1 for 2 1 1 2 2 − a a as numerator of both fractions B1 for 1 as part of numerator in both brackets Accept equivalent forms for 2 2 2 1 1 1 1 2 2 2 2 2 2 − − − a a a a b b

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 5 of 8 Question Answer Marks Partial Marks 6(a) a = b and ( 1) 2 − × a a ( 1) 2 − a a 1 6(b)(i) Correct diagram with ruled lines 1 6(b)(ii) 7 and 9 One point has more than two lines crossing oe 2 B1 for both values and B1 for reason 6(c)(i) Correct value and correct explanation 1 6(c)(ii) 19 1 Communication: seen in two of the following questions 1 2(b) for showing 3 second differences = 1 or simultaneous equations/formula 3(b) for showing 3 second differences = 3 or simultaneous equations 4 for showing 3 second differences = 6 or simultaneous equations 6(c)(ii) for sketch or trial and improvement with at least 3 trials or simplifying to (a(a – 1))2 = 116 964 oe or a(a – 1) = 342 oe

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 6 of 8 Question Answer Marks Partial Marks B MODELLING VALUE OF A CAR 7(a) 4000 2000 1000 500 250 1 7(b) Correct curve 3 B2FT for 5, 6 or 7 correct plots or B1FT for 3 or 4 correct plots 7(c) 0.5x oe 1 7(d) 15.63 1 8(a) 7 correct plots 1 8(b) [John’s =] 6720 1 Anna’s value in the table written or referred to 1 Strict FT their 500 from 7(a)

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 7 of 8 Question Answer Marks Partial Marks 9(a) Correct sketch 2 B1 for correct shape 9(b) Not well oe and [Eddie’s] model [usually] gives lower/different values 1 Comment must imply differences between corresponding values in the two models 10(a) [0].4 1 10(b) Correct sketch 1 Curve always concave upwards starting at (0, 100) and always above horizontal axis and reaching x = 9.5 10(c) Correct comment that recognises good fit for earlier years or poor fit as years increase or overall poor fit as percentages [usually] below/different 1 11(a) V = 1 ( ) + C their a x oe Final answer 1 11(b) 3 years 8 months or 3 years 9 months 2 B1 for 3.65 to 3.85 seen or 44 months or 45 months seen

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2019 © UCLES 2019 Page 8 of 8 Question Answer Marks Partial Marks Communication: seen in three of the following questions 2 C1 for 1 opportunity seen 7(a) for division by 2 at least once 7(d) for repeated division by 2 or for 3 values 125, 62.5, 31.25 or 16 000 × 10 1 2 oe 8(b) 16 000 × 42/100 oe 10(a) for correct substitution of 72 [and 1] and next step correct in rearrangement 11(b) for sketch or correct substitution of 6500 in their 11(a)and next step correct in rearrangement