Cambridge IGCSE Mathematics - International 0607 — 2023 Oct/Nov Paper 6 · Variant 2

This document has 16 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 Investigation and Modelling (Extended) October/November 2023 1 hour 40 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer both part A (Questions 1 to 4) and part B (Questions 5 to 7). ● 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 [ ]. * 8 3 3 4 8 2 7 3 8 3 * DC (CJ/FC) 318330/2 © UCLES 2023

2 0607/62/O/N/23 © UCLES 2023 The investigation starts on page 3.

3 0607/62/O/N/23 © UCLES 2023 [Turn over Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 4) CLOCK HANDS (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at the angle between the hands of a clock at different times of day. You should not measure angles from the clock diagrams. In this investigation: • the hour hand is labelled H • the minute hand is labelled M • the hands of the clock rotate clockwise in the direction shown • the clockwise angle between the two hands is shown on the clock. 1 (a) The clock shows the time 1.00 am. In one hour, hand H rotates clockwise from one number to the next number. For example, from 1.00 am to 2.00 am hand H rotates from 1 to 2. Show that hand H rotates 0.5° in one minute. [2] (b) In one hour, hand M rotates through a full circle. Show that hand M rotates 6° in one minute. [1] cl o c k w is e M H 12 6 7 8 1 11 2 10 3 9 4 5

4 0607/62/O/N/23 © UCLES 2023 2 m is the number of minutes after the last hour. In this question the last hour is 1.00 am. Examples At 1.10 am, m = 10. At 1.45 am, m = 45. (a) This clock shows the time 1.10 am. Show that the clockwise angle from hand H to hand M at 1.10 am is 25°. [2] (b) Complete the table. You may use the clock diagrams to help you. Number of minutes after the last hour (m) Angle rotated since 1.00 am in degrees Clockwise angle between the hands in degrees Hand H angle Hand M angle 6 7 8.5 8 9 10 25 12 6 7 8 1 11 2 10 3 9 4 5 12 6 7 8 1 11 2 10 3 9 4 5 12 6 7 8 1 11 2 10 3 9 4 5 [4] M H 12 6 7 8 1 11 2 10 3 9 4 5

5 0607/62/O/N/23 © UCLES 2023 [Turn over (c) Find an expression, in terms of m, for the clockwise angle between the hands when the last hour is 1.00 am. … [2] (d) Find how many minutes and seconds after 1.00 am the clockwise angle is 270°. Give your answer correct to the nearest second. … minutes … seconds [4]

6 0607/62/O/N/23 © UCLES 2023 3 In this question the last hour is 2.00 am. (a) This clock shows the time 2.15 am. Show that the clockwise angle between the hands is 22.5°. [1] (b) Complete the table. You may use the clock diagrams to help you. Number of minutes after the last hour (m) Clockwise angle between the hands in degrees 15 22.5 16 17 18 19 12 6 7 8 1 11 2 10 3 9 4 5 12 6 7 8 1 11 2 10 3 9 4 5 12 6 7 8 1 11 2 10 3 9 4 5 [2] (c) Find an expression, in terms of m, for the clockwise angle between the hands when the last hour is 2.00 am. … [2] M H 12 6 7 8 1 11 2 10 3 9 4 5

7 0607/62/O/N/23 © UCLES 2023 [Turn over 4 (a) h is the number of hours in the time. m is the number of minutes after the last hour. Examples At 1.30 am, h = 1 and m = 30. At 8.45 am, h = 8 and m = 45. Complete the table using expressions of the form am + b. Use your expressions from Question 2(c) and Question 3(c). Number of hours in the time (h) Clockwise angle in degrees between the hands m minutes after h Question 2(c) 1 Question 3(c) 2 3 4 [1] (b) Find an expression, in terms of m and h, for the clockwise angle between the hands. … [2]

8 0607/62/O/N/23 © UCLES 2023 (c) You may use the clock diagrams to help you in this part. (i) Use your expression from part (b) to find the two angles between the hands at 10.12 am. … and … [3] (ii) There are two times between 7.00 am and 8.00 am when an angle between the hands is 100°. Find these times correct to the nearest minute. … and … [4] 12 6 7 8 1 11 2 10 3 9 4 5 12 6 7 8 1 11 2 10 3 9 4 5

9 0607/62/ O/ N/23 © UCLES 2023 [Turn over B MODELLING (QUESTIONS 5 TO 7) ARCHES (30 marks) You are advised to spend no more than 50 minutes on this part. Circumference, C, of circle, radius r. r C r 2 = This task looks at models for the lengths of arches. An arch is the curved part of a tunnel or bridge. Engineers use the dimensions of an arch to calculate its strength. In this task, each arch has: • width w metres • height h metres • curved length l metres. 5 Semicircular arch model w h This arch is a semicircle. The width is 8 m. (a) Write down the height of the arch. … [2] (b) Find the curved length of the arch. Give your answer correct to the nearest centimetre. … [2] Content removed due to copyright restrictions.

10 0607/62/O/N/23 © UCLES 2023 6 Segmental arch model w h NOT TO SCALE This arch is an arc of a circle with radius r metres. The width of the arch is 8 metres. (a) The formula for the radius of the circle is r h h w 8 4 2 2 = + . (i) Show that r h h 2 8 = + . [2] (ii) On the diagram, sketch the graph of r for h 0 8 1 G . h r 10 0 0 8 [2] (iii) Find the height that gives the minimum radius. … [1]

11 0607/62/O/N/23 © UCLES 2023 [Turn over (b) (i) The height of the arch is 1 metre. Find the radius of the arch. … [1] (ii) r O 8 θ NOT TO SCALE The angle at the centre of the circle that forms the arch is θ. Find the value of θ correct to the nearest degree. … [3] (iii) Find the curved length of the arch. … [2]

12 0607/62/O/N/23 © UCLES 2023 7 Lancet arch model w h This arch is made using two equal arcs. r O O h w The equal arcs are parts of the circumferences of two identical circles both of radius r metres. The base of the arch is on the line joining the centres of the two circles. The height is on the line of symmetry of the arch. (a) Use Pythagoras’ Theorem to show that the model for r in terms of h and w is r w h w 4 2 = + . [4]

13 0607/62/O/N/23 © UCLES 2023 [Turn over (b) The width of the arch is 8 metres. (i) Show that r h 8 2 2 = + . [1] (ii) On the diagram in Question 6(a)(ii), sketch the graph of r for h 0 8 1 G . [2] (iii) Find the value of h and the value of r when the graphs from Question 6(a)(ii) and Question 7(b)(ii) intersect. Explain what your answers show about the shapes of the two arches at this point. h = … r = … … [3]

14 0607/62/O/N/23 © UCLES 2023 (c) The width of the lancet arch is 8 metres. The radius is 10 metres. (i) Find its height. … [1] (ii) 10 O O h Use trigonometry to help you find the curved length of the arch. … [4]

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16 0607/62/O/N/23 © UCLES 2023 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. BLANK PAGE

This document consists of 8 printed pages. © UCLES 2023 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) October/November 2023 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 2023 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 2023 © UCLES 2023 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 2023 © UCLES 2023 Page 3 of 8 Mathematics-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, non-integer 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 or sign 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 A or B 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

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 4 of 8 Question Answer Marks Partial Marks 1(a) 360 12 60  oe leading to 0.5 2 B1 for 360 12 or for 360 60 or for 30 60 1(b) 360 60 leading to 6 oe or 30 5 leading to 6 oe 1 2(a) 10 𝗑 6 – 10 𝗑 0.5 – 30 leading to 25 i.e. 10 × 6 – (10 × 0.5 + 30) leading to 25 or 30 – 10 𝗑 0.5 leading to 25 or 10 𝗑 6 – 70 𝗑 0.5 leading to 25 or 30 – 30 6 leading to 25 2 B1 for 10 𝗑 0.5 or 10 𝗑 6 or 70 𝗑 0.5 or 30 6 or 30 – 5 = 25 2(b) 3 correct from: three differences of 0.5 seen three differences of 6 seen three differences of 5.5 seen at least one calculation for H e.g. for top row 6 × 0.5 at least one calculation for M e.g. for top row 6 × 6 at least one calculation for final column e.g. for top row 36 – 30 – 3 C2 C1 for 1 correct (m) H M angle 6 3 36 3 7 3.5 42 8.5 8 4 48 14 9 4.5 54 19.5 10 5 60 25 2 B1 for one column correct 2(c) 5.5m – 30 oe 2 B1 for 5.5m oe e.g. 6m – 2 m or km – 30 (k ≠ 0)

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 5 of 8 Question Answer Marks Partial Marks 2(d) 5.5m – 30 = 270 C1 FT their (5.5m – 30) Rearrange equation to get [m =] 300 5.5 oe 1 FT their equation (decimal part of their m) 𝗑 60 C1 54 minutes 33 seconds 1 3(a) 15 𝗑 6 – 15 𝗑 0.5 – 60 oe leading to 22.5 or 30 – 15 𝗑 0.5 oe leading to 22.5 or 15 𝗑 6 – 135  0.5 leading to 22.5 1 3(b) One relevant calculation C1 (m) angle 15 22.5 16 28 17 33.5 18 39 19 44.5 1 3(c) 5.5m – 60 oe 2 B1 for 5.5m oe or km – 60 (k ≠ 0) 4(a) (h) angle 1 5.5m – 30 2 5.5m – 60 3 5.5m – 90 4 5.5m – 120 1 4(b) 5.5m – 30h oe 1 two differences of 30 for subtracted term seen or two products for the subtracted term C1 4(c)(i) substitution of m = 12 and h = 10 into 5.5m – 30h C1 FT their (5.5m – 30h) 234 and 126 2 B1 for each

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 6 of 8 Question Answer Marks Partial Marks 4(c)(ii) 5.5m – 30h = [+ or/and –] 100 C1 FT their (5.5m – 30h) substitution of h = 7 into 5.5m – 30h [= [+ or –]100] C1 FT their (5.5m – 30h) 7.20 [am] and 7.56 [am] 2 B1 for one or B1 for m = 310 5.5 oe or m = 110 5.5 oe 5(a) 4 1 metres or m C1 5(b) π 8 2  or 2 π 4 2  oe C1 12.57 m oe 1 6(a)(i) Substitution of w = 8 and splitting into two fractions leading to 8 2 h r h = + 2 B1 for one correct stage 6(a)(ii) Correct sketch 2 B1 if approximately correct shape but with minimum outside tolerance 6(a)(iii) 4 m 1 6(b)(i) 8.5 m 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 7 of 8 Question Answer Marks Partial Marks 6(b)(ii) Correct statement to find θ e.g. 1 4 sin 2 8.5  −   =      or 2 2 2 8.5 8.5 8 cos 2 8.5 8.5  + − =   Correct statement to find 2  followed by  2 in working e.g. 4 sin 2 8.5 = or 1 4 sin 2 8.5  − = etc. or 2 2 2 8.5 7.5 4 cos 2 2 8.5 7.5  + − =   or 82 = 8.52 + 8.52 – 2  8.5  8.5 cos  followed by 80.5 144.5 4 sin90 sin 2 8.5   = C2 FT their 8.5 C1 for incorrect statement using θ instead of 2 e.g. 4 sin 8.5 = or correct working without θ or 82 = 8.52 + 8.52 – 2  8.5  8.5 cos  or 42 = 8.52 + 7.52 – 2  8.5  7.5 cos 2  or correct statement to find 2  not followed by  2 in working sin sin90 2 4 8.5  = oe 56° 1 6(b)(iii) 56 2 8.5 360     oe C1 FT their 56 FT their 8.5 8.31 metres 1 7(a) 2 2 2 2 w r h r   = + −     oe M2 M1 for 2 w r − seen oe 2 2 2 2 4 w w w r r r − − + M1 Expansion of brackets 2 2 4 w rw h = + oe leading to 2 4 h w r w = + A1 7(b)(i) Substitution of w = 8 leading to 2 2 8 h r = + 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 8 of 8 Question Answer Marks Partial Marks 7(b)(ii) Correct sketch 1 r–axis intercept labelled at 2 C1 7(b)(iii) h = 4 r = 4 2 B1 for each Both semi-circular arches oe 1 7(c)(i) 8 m 1 7(c)(ii) 8 sin 10 = or 6 cos 10 = or 8 tan 6 = or sin sin90 8 10 = oe C1 FT their 8 53.1° 1 53.1 2 π 10 360  [ 2] oe C1 FT their 53.1 18.5 m 1