Cambridge IGCSE Mathematics - International 0607 — 2022 May/June Paper 6 · Variant 3

This document has 16 pages. [Turn over Cambridge IGCSE™ DC (CJ/SG) 305621/3 © UCLES 2022 * 1 8 8 2 8 2 7 9 7 1 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 Investigation and Modelling (Extended) May/June 2022 1 hour 40 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer both part A (Questions 1 to 7) and part B (Questions 8 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 [ ].

2 0607/63/M/J/22 © UCLES 2022 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 7) CIRCLES AND REGIONS (30 marks) You are advised to spend no more than 50 minutes on this part. This is an investigation into the number of regions formed by drawing lines on a circle. 1 Radii The diagrams show the number of regions inside a circle when 1 radius and 2 radii are drawn. The regions inside the circle are numbered. 1 1 radius 1 2 2 radii (a) Complete the table. Number of radii Number of regions 1 1 2 2 3 4 5 6 [1] (b) Write a formula, in terms of n, for the number of regions, R, when there are n radii. … [1]

3 0607/63/M/J/22 © UCLES 2022 [Turn over 2 Diameters The diagrams show the number of regions inside a circle when 1 diameter and 2 diameters are drawn. 1 diameter 1 2 2 diameters 1 2 3 4 (a) Complete the table for 3, 4 and 5 diameters. You may use the empty circle to help you. Number of diameters Number of regions 1 2 2 4 3 4 5 [2] (b) Write a formula, in terms of n, for the number of regions, R, when there are n diameters. … [1]

4 0607/63/M/J/22 © UCLES 2022 3 Chords In this investigation: • each chord must cut every other chord • only two chords may intersect at any point. The diagrams show the number of regions inside a circle when 1 chord, 2 chords and 3 chords are drawn. 3 chords 2 chords 1 chord 1 1 2 3 4 4 7 5 6 3 1 2 2 (a) Count the number of regions in the circle when 4 chords are drawn. … [1]

5 0607/63/M/J/22 © UCLES 2022 [Turn over (b) Complete this table. You may use the empty circle to help you. Number of chords Number of regions 1 2 2 4 3 7 4 5 6 22 [1] (c) Find a formula, in terms of n, for the number of regions, R, when there are n intersecting chords. … [4]

6 0607/63/M/J/22 © UCLES 2022 4 Tangents A region can be inside or outside the circle when the lines are tangents. These two diagrams both show a circle with 2 tangents and the regions numbered. The maximum number of regions for a circle with 2 tangents is 6. 3 2 4 1 5 2 1 3 5 4 6 (a) Give a reason why the first diagram does not have the maximum number of regions with 2 tangents. … … [1] (b) Use this diagram to find the maximum number of regions when there are 3 tangents. … [1]

7 0607/63/M/J/22 © UCLES 2022 [Turn over (c) Draw a fourth tangent on the diagram below to find the maximum number of regions. Complete the table. Number of tangents Maximum number of regions 1 3 2 6 3 4 5 21 [2] (d) This is a formula for the maximum number of regions, R, when there are n tangents. R n bn 2 1 1 2 = + + Find the value of b. … [2]

8 0607/63/M/J/22 © UCLES 2022 5 Secants A secant is a straight line that intersects a circle at two points and extends outside the circle. In this investigation: • each secant must cut every other secant • only 2 secants may intersect at any point • secants must not intersect on the circumference of the circle. secant The diagram shows the number of regions with 2 secants drawn on a circle. 2 6 8 7 5 4 3 1 (a) Find the number of regions when there are 3 secants. Complete the table. Number of secants Number of regions 1 4 2 8 3 4 19 5 26 [2]

9 0607/63/M/J/22 © UCLES 2022 [Turn over (b) This is a formula for the number of regions, R, when there are n secants. R n bn c 2 1 2 = + + Find the value of b and the value of c. b = … c = … [4]

10 0607/63/M/J/22 © UCLES 2022 6 Tangents are drawn on a circle to give the maximum number of regions. There are 1225 regions. Find the number of tangents. … [3] 7 There are two circles. The first circle has chords drawn on it. The second circle has secants drawn on it. The number of chords on the first circle is the same as the number of secants on the second circle. Each circle has the maximum number of regions. One circle has 60 more regions than the other. (a) Find the number of straight lines on each diagram. … [2] (b) Find the larger number of regions. … [2]

11 0607/63/M/J/22 © UCLES 2022 [Turn over B MODELLING (QUESTIONS 8 TO 11) AIRPORT RUNWAY (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at the factors that affect the decision to build a second runway at an airport. The factors are: • the number of seconds a plane waits over the airport before it can start to land • the number of seconds it then takes to land. A plane cannot begin to land until the runway is free but must wait over the airport. As soon as the runway is free the plane begins to land. The number of seconds between one plane and the next plane arriving over the airport is called the inter-arrival time. The number of seconds from when a plane begins to land and when it stops is called the landing time. 8 This table shows the data for the first 5 planes arriving at the airport for the 1080 seconds after 8 am on day 1. For example: Plane B arrives 120 seconds after plane A. Plane A has not ended its landing. Plane B starts its landing 180 seconds after 8 am, as soon as plane A has ended its landing. Plane Inter-arrival time (seconds) Arrival over airport (seconds after 8 am) Start of landing (seconds after 8 am) Landing time (seconds) End of landing (seconds after 8 am) Seconds waiting to land A 30 30 150 180 0 B 120 150 180 110 290 30 C 360 510 510 100 610 0 D 25 535 610 280 75 E 60 44 934 (a) Complete the table. [5] (b) A plane uses the runway for the whole of its landing time. Calculate the total time that the runway was not used during these 1080 seconds. … [2]

12 0607/63/M/J/22 © UCLES 2022 9 To decide if a second runway should be built, more data is needed. The table shows the data for 180 planes. All values are given correct to the nearest integer. (a) Complete the table. Inter-arrival time (t seconds) Number of planes Percentage of planes Inter-arrival time (t seconds) Cumulative percentage of planes (p) 0 t 1 G 60 42 23 t G 60 23 60 t 1 G 120 34 19 t G 120 42 120 t 1 G 180 29 t G 180 180 t 1 G 240 23 t G 240 71 240 t 1 G 300 16 t G 300 80 300 t 1 G 360 11 6 t G 360 86 360 t 1 G 420 11 6 t G 420 420 t 1 G 480 7 4 t G 480 480 t 1 G 540 4 2 t G 540 540 t 1 G 600 2 1 t G 600 99 600 t 1 G 660 0 0 t G 660 99 660 t 1 G 720 0 0 t G 720 99 720 t 1 G 780 1 1 t G 780 100 780 t 1 G 840 0 0 t G 840 100 840 t 1 G 900 0 0 t G 900 100 [3]

13 0607/63/M/J/22 © UCLES 2022 [Turn over (b) On the grid below, complete the cumulative percentage curve. p t 0 0 20 40 60 80 100 100 200 300 400 500 Time (seconds) 600 700 800 900 1000 Cumulative percentage [2] (c) Use the graph to estimate the inter-arrival time for a cumulative percentage of 50. … [1]

14 0607/63/M/J/22 © UCLES 2022 10 This is a model for the cumulative percentage, p, in terms of the inter-arrival time, t. ( ) p k t a 1 3 = - where a and k are constants (a) Use the points (60, 23) and (360, 86) to write down two equations in terms of a and k. … … [1] (b) Show that . a a 52 3 60 360 = - - , where 52.3 is correct to 1 decimal place. [2] (c) Solve the equation in part (b) to show that a 54 = , correct to the nearest integer. [3]

15 0607/63/M/J/22 © UCLES 2022 [Turn over (d) Find the value of k, correct to the nearest integer, and complete the model. … ( …) p t 3 1 = - [2] (e) Use the model to find the inter-arrival time for a cumulative percentage of 50. … [3] (f) Sketch the model on the axes in Question 9(b). [2] (g) Comment on the validity of this model. … … [1] Question 11 is printed on the next page.

16 0607/63/M/J/22 © UCLES 2022 11 (a) Use the model to find the percentage of planes that arrived over the airport within 120 seconds of the previous plane. … [1] (b) The table shows information about landing times for the 180 planes. All values are given correct to the nearest integer. Landing time (t seconds) Number of planes Percentage of planes Landing time (t seconds) Cumulative percentage of planes (p) 0 t 1 G 60 2 1 t G 60 1 60 t 1 G 120 7 4 t G 120 5 120 t 1 G 180 11 6 t G 180 11 180 t 1 G 240 15 8 t G 240 19 240 t 1 G 300 20 11 t G 300 30 300 t 1 G 360 34 19 t G 360 49 360 t 1 G 420 42 23 t G 420 72 420 t 1 G 480 30 17 t G 480 89 480 t 1 G 540 18 10 t G 540 99 540 t 1 G 600 1 1 t G 600 100 Find the percentage of planes where the landing time is more than 120 seconds. … [1] (c) Based on your answers to part (a) and part (b), should a second runway be built at the airport? Give a reason for your answer. … … [1] 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 Cambridge Assessment. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which is a department of the University of Cambridge.

This document consists of 8 printed pages. © UCLES 2022 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2022 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 May/June 2022 series for most Cambridge IGCSE, Cambridge International A and AS Level and Cambridge Pre-U components, and some Cambridge O Level components.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 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.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 3 of 8 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 2022 © UCLES 2022 Page 4 of 8 Question Answer Marks Partial Marks 1(a) 3, 4, 5, 6 in correct rows 1 1(b) R = n 1 2(a) Circle with 3, 4 or 5 diameters drawn C1 6, 8, 10 in correct rows 1 2(b) R = 2n 1 3(a) 11 1 3(b) 16 in row 5 1 3(c) Three second differences of 1 seen AND coefficient of n2 equal to 1 2 seen or use of quadratic expression seen C1 Derivation of coefficient of n equal to 1 2 seen C1 Derivation of constant equal to 1 seen C1 2 1 1 1 2 2 R n n    1 3(c) alternative 1 Three second differences of 1 seen AND coefficient of n2 equal to 1 2 seen C1 [½ n2] 0.5 2 4 .5 8 12.5 C1 [leaves ] 1.5 2 2.5 3 3.5 One difference of 0.5 seen (b =) 0.5 C1 2 1 1 1 2 2 R n n    1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 5 of 8 Question Answer Marks Partial Marks 3(c) 3(c) alternative 2 Three second differences of 1 seen AND coefficient of n2 equal to 1 2 seen C1 1 2 4 7 11 16 22 1 2 3 4 5 6 1 1 1 1 1 C1 a + b = 1 indicated C1 2 1 1 1 2 2 R n n    1 4(a) [tangents] do not intersect or [tangents are] parallel 1 4(b) 10 1 4(c) 4th tangent drawn that intersects the other 3 tangents 1 15 (on row 4 of table) 1 4(d) Equation with correct substitution for R and n C1 1.5 oe 1 5(a) 3 secants intersecting each other C1 13 1 5(b) Two equations with correct substitution for R and n 1 Correct method to eliminate either b or c C1 (b =) 2.5 (c =) 1 2 B1 for each 5(b) alternative 1 [½ n2 ] 0.5 2 4 .5 8 12.5 [leaves] 3.5 6 8.5 11 13.5 1 One difference of 2.5 seen C1 (b =) 2.5 (c =) 1 2 B1 for each

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 6 of 8 Question Answer Marks Partial Marks 5(b) 5(b) alternative 2 1 4 8 13 19 26 3 4 5 6 7 1 a + b = 3 indicated C1 (b =) 2.5 (c =) 1 2 B1 for each 6 Sketch of the quadratic or correct substitution and rearrangement to = 0 C1 FT their 4(d) Correct intersection of curve and straight line marked on sketch or correct factorisation or correct substitution into quadratic formula C1 48 1 7(a) Their 5(b) – their 3(c) = 60 oe Or list of results showing differences C1 30 1 7(b) Correct substitution of their 30 into 5(b) or 3(c) C1 FT their 5(b) or their 3(c) 526 1 8(a) Correct calculation for a correct figure in table C1 Plan inter arrive start land end wait D 25 535 610 280 890 75 E 60 595 890 44 934 295 4 B1 for each cell FT for 295, their 890 – their 595 8(b) 1080 – (150 + 110 + 100 + 280 + 44) oe C1 396 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 7 of 8 Question Answer Marks Partial Marks 9(a) Inter Num % Cum % 60<t⩽120 34 19 t ⩽ 120 42 120<t⩽180 29 16 t ⩽ 180 58 180<t⩽240 23 13 t ⩽ 240 71 240<t⩽300 16 9 t ⩽ 300 80 300<t⩽360 11 6 t ⩽ 360 86 360<t⩽420 11 6 t ⩽ 420 92 420<t⩽480 7 4 t ⩽ 480 96 480<t⩽540 4 2 t ⩽ 540 98 540<t⩽600 2 1 t ⩽ 600 99 3 B1 for 16 B2 for other 6 cells correct Or B1 for 3, 4 or 5 other cells correct FT their 16 for 58 FT their 58 for 13 9(b) Points correctly plotted 1 FT their points Correct curve 1 FT their plotted points 9(c) Between 140 and 160 1 10(a)   1 3 23 60 k a   isw   1 3 86 360 k a   isw 1 10(b)     1 3 1 3 360 86 23 60 a a    1 3 86 360 23 60 a a         1 10(c)   52.3 60 360 a a    1 Correctly isolating a C1 54.1 […] or 54.2 leading to 54 1 10(d) Correct substitution of 54 into one of the equations in part (a) C1 p = …13…  1 3 54 t   1 FT

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 8 of 8 Question Answer Marks Partial Marks 10(e)   1 3 50 13 54 their t   or correct sketch of model 1 Correct first step or correct sketch with line at 50 C1 111 1 FT their 13 10(f) Correct sketch 2 FT their model from 10(d) B1 for correct shape B1 for position 10(g) Valid up to t = approx. 500 or invalid after t = approx. 500 1 FT their model 11(a) 52 to 53 1 FT their model from 10(d) 11(b) 95 1 11(c) Yes, with appropriate comment e.g. approx. half the planes have to wait to land 1