Cambridge IGCSE Mathematics - International 0607 — 2022 Oct/Nov Paper 6 · Variant 3

This document has 16 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ DC (KN/SG) 303075/2 © UCLES 2022 * 7 6 5 7 7 1 4 8 9 0 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 Investigation and Modelling (Extended) October/November 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 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 [ ].

2 0607/63/O/N/22 © UCLES 2022 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 4) REMAINING SHAPES (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at the area of the remaining shape when part of a larger shape is removed. The side difference is the difference between the lengths of the sides of the two shapes. In this investigation all lengths are in centimetres. 1 A square is removed from a larger square. The side difference is 1. A square of side 2 is removed from a larger square of side 3. The area of the remaining shape is 5. A square of side 3 is removed from a larger square of side 4. This is the remaining shape. (a) On this grid, draw the remaining shape for a larger square of side 5. [1]

3 0607/63/O/N/22 © UCLES 2022 [Turn over (b) Complete the table. Side of larger square Area of remaining shape 2 3 5 4 5 6 [3] (c) Find an expression, in terms of n, for the area of the remaining shape when the larger square has side n. … [2] (d) The area of a remaining shape is 381. Work out the length of the side of the larger square. … [2]

4 0607/63/O/N/22 © UCLES 2022 2 The side length of the square removed is now 2 less than the side length of the larger square. The side difference is 2. A square of side 2 is removed from a larger square of side 4. The area of the remaining shape is 12. (a) Complete the table. You may use the grid below to help you. Side of larger square Area of remaining shape 3 4 12 5 6 [3] (b) Complete the expression for the area of the remaining shape when the larger square has side n and the side difference is 2. 4n … [1]

5 0607/63/O/N/22 © UCLES 2022 [Turn over 3 (a) Complete the table using expressions in the form an b + , where a and b are integers. Use your expressions from Question 1(c) and Question 2(b). You may use the grid below to help you. Side difference (d) Area of remaining shape for larger square of side n 1 2 3 4 [4] (b) Find an expression, in terms of n and d, for the area of the remaining shape when the side difference is d. … [2] (c) The area of a remaining shape is 343. The side difference is 7. Find the length of the side of the larger square. … [2]

6 0607/63/O/N/22 © UCLES 2022 4 Area sin bc A 2 1 = A a C b B c This question looks at the area of a remaining shape for an equilateral triangle. (a) Show that the area of an equilateral triangle with side of length n is n 4 3 2 . [1] (b) An equilateral triangle is removed from a larger equilateral triangle. The side difference is 1. An equilateral triangle with a side of length 3 is removed from a larger equilateral triangle with a side of length 4. 3 4 Show that the area of the remaining shape is 4 7 3. [1]

7 0607/63/O/N/22 © UCLES 2022 [Turn over (c) Find the area of the remaining shape for a larger equilateral triangle with a side of length 5 and a side difference of 1. Give your answer in the form k 4 3. … [2]

8 0607/63/O/N/22 © UCLES 2022 (d) Complete the table. Side of larger triangle Area of remaining shape 2 3 4 4 7 3 5 [3] (e) Find an expression, in terms of n and d, for the area of the remaining shape for a larger equilateral triangle with side of length n when the side difference is d. Give an exact answer in its simplest form. … [3]

9 0607/63/O/N/22 © UCLES 2022 [Turn over B MODELLING (QUESTIONS 5 TO 7) ESCALATORS (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at going up and down on escalators. 5 (a) Every step of an escalator travels at a speed of 1.8 km/h. Show that the distance a step moves in 1 second is 0.5 m. [2] (b) The length of the escalator is 40 m. Find the time, in seconds, for a step to travel the length of the escalator. … [2]

10 0607/63/O/N/22 © UCLES 2022 6 Matt travels up a different escalator. The length of this escalator is 24 metres. (a) (i) The escalator is not moving. It takes Matt 50 seconds to walk up the length of the escalator. Find his speed. … [2] (ii) The escalator is now moving. When Matt stands on the escalator it takes him 30 seconds to move up the length of the escalator. Matt now walks up the moving escalator at the same speed as in part (a)(i). Show that his speed walking up the moving escalator is 1.28 m/s. [2] (iii) Find how long it will take him to walk up the length of the moving escalator. … [2]

11 0607/63/O/N/22 © UCLES 2022 [Turn over (b) (i) Matt walks down the escalator when it is not moving. It takes him 24 seconds to walk down the length of the escalator. The escalator starts moving upwards at the same speed as in part (a). Matt now walks down the moving escalator. Show that his speed walking down the moving escalator is 0.2 m/s. [2] (ii) Find how long it will take Matt to walk down the length of the escalator when it is moving upwards at the same speed as before. … [2]

12 0607/63/O/N/22 © UCLES 2022 7 Another escalator has a length of 6.2 m. The escalator travels upwards at a speed of v metres per second. (a) With the escalator travelling upwards, Matt walks up the escalator and then immediately turns around and walks down the escalator. His walking speed is 0.8 m/s. The total time for Matt to walk up and down the moving escalator is T seconds. (i) Give reasons why the model for T is . . . . T v v 0 8 6 2 0 8 6 2 = + + - . … … … [3] (ii) Show that the model simplifies to . . T v 0 64 9 92 2 = - . [3] (iii) When the speed of the escalator is 0.4 m/s, find how long it takes Matt to walk up and down the escalator. … [1]

13 0607/63/O/N/22 © UCLES 2022 (b) Matt continues to walk at 0.8 m/s. He walks up the moving escalator, waits for 10 seconds, then walks down the escalator. (i) Change the model in part (a)(ii) to find a new model for T. You do not need to simplify your model. … [1] (ii) Sketch the graph of T for v 0 5 G G . T – 50 150 0 5 v [4] (iii) The total time for Matt to walk up, wait, and walk down the moving escalator is 138 seconds. Find the speed of the escalator. … [2] (iv) Give a reason why the model is not valid when v 5 = . … [1] (v) For what range of values of v is the model valid? … [1]

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16 0607/63/O/N/22 © UCLES 2022 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 2022 [Turn over Cambridge IGCSE™ MATHEMATICS 0607/63 Paper 6 Extended October/November 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 October/November 2022 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 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 October/November 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 October/November 2022 © UCLES 2022 Page 4 of 8 Question Answer Marks Partial Marks 1(a) 1 1(b) Side of larger square Area of remaining shape 2 3 3 5 4 7 5 9 6 11 3 B1 for 3 in first row B1 for 7 and 9 B1 for 11 in last row 1(c) 2n – 1 oe 2 B1 for 2n or n2 – (n – k)² (k ≠ 0) (k ≠ 1) 1(d) (2n – 1) = 381 C1 FT their (2n – 1) 191 1 2(a) One correct diagram or 32 – 12 or 52 – 32 or 62 – 42 oe C1 Side of larger square Area of remaining shape 3 8 4 12 5 16 6 20 2 B1 for 2 correct 2(b) – 4 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 5 of 8 Question Answer Marks Partial Marks 3(a) One correct diagram with side difference of 3 or 4 or 15, 21, 27 or 24, 32, 48 or n² – (n – 3)² or n² – (n – 4)² C1 Side difference (d) Area of remaining shape for larger square of side n 1 2n – 1 2 4n – 4 3 6n – 9 4 8n – 16 3 B2 for 6n – 9 or 8n – 16 or B1 for 6n + k and 8n + k for any k 3(b) 2dn – d² oe or n2 – (n – d)2 isw 2 B1 for 2dn – a where a ≠ 0 or for bn – d² where b ≠ 0 or for (n – d)2 seen 3(c) Substitution of d = 7 into (2dn – d²) = 343 or substitution of d = 7 into n2 – (n – d)2 = 343 C1 FT their answer to 3(b) in terms of n and d 28 1 4(a) 2 1 1 3 sin60 2 2 2  = n n n 1 4(b) 2 2 4 3 3 3 4 4 − leading to 7 3 4 1 4(c) 2 2 5 3 4 3 4 4 − oe C1 9 3 4 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 6 of 8 Question Answer Marks Partial Marks 4(d) Correct calculation for either side 2 or 3 C1 Side of larger triangle Area of remaining shape 2 3 3 4 3 5 3 4 4 7 3 4 5 9 3 4 2 B1 for 3 3 4 or 5 3 4 4(e) 2 2 3 ( ) 3 4 4 − − n n d C1 3 4       (n² – (n² – nd – nd + d2)) or ( ) ( ) ( ) ( ) 3 4   − − + −     n n d n n d C1 3 (2 ²) 4 − dn d or 3 (2 ) 4 − d n d isw oe 1 5(a) 1.8 1000 60 60   2 M1 for  1000 or ÷ (60  60) 5(b) 40 0.5 or   0.04 3600 1.8  oe or 0.5  60 (30m) in 60 secs and 30 ÷ 3 (10m) in 20 secs oe C1 80 [secs] 1 6(a)(i) 24 50 C1 0.48 [m/s] 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 7 of 8 Question Answer Marks Partial Marks 6(a)(ii) 24 30 oe 1 0.8 + 0.48 leading to 1.28 1 6(a)(iii) 24 1.28 oe C1 18.75 [secs] 1 6(b)(i) 1 – 0.8 oe leading to 0.2 2 B1 for 24 24 or 1 seen 6(b)(ii) 120 1 secs C1 7(a)(i) Time = distance ÷ speed 1 6.2 0.8+ v is time up or 0.8 + v is speed up 1 6.2 0.8−v is time down or 0.8 – v is speed down 1 7(a)(ii) Numerator 6.2(0.8 ) 6.2(0.8 ) − + + v v Denominator (0.8 )(0.8 ) + − v v 2 B1 for each   2 4.96 6.2 4.96 6.2 0.64 0.8 0.8 − + + = + − − v v T v v v leading to 9.92 0.64 ² −v 1 7(a)(iii) 20.7 [secs] 1 7(b)(i) 9.92 10 0.64 ² = + − T v 1 7(b)(ii) 2 B1 Two branches of approx. correct shape and approx. correct position B1 dep for not overlapping nor curling back nor point of inflexion v–axis intercept labelled at 1.28 C1 T–axis intercept labelled at 25.5 C1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 8 of 8 Question Answer Marks Partial Marks 7(b)(iii) horizontal line at T = 138 or 138 = 9.92 10 0.64 ² + −v C1 FT their model in (b)(i) 0.75 [m/s] 1 7(b)(iv) speed of escalator is larger than Matt’s speed or time is less than 10 seconds 1 7(b)(v) [0 ≤ ]v < 0.8 1