Cambridge IGCSE Mathematics - International 0607 — 2023 Oct/Nov Paper 6 · Variant 3

This document has 16 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ DC (CE/SG) 318331/2 © UCLES 2023 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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 [ ]. * 7 6 8 9 7 9 6 7 6 0 *

2 0607/63/O/N/23 © UCLES 2023 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 4) GEOMETRIC SEQUENCES (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at sequences where the rule to get from one term to the next is a multiplication. These are called geometric sequences. 1 (a) These are the first four terms of a geometric sequence. 4 16 64 256 (i) Work out the next term. … [1] (ii) Work out the 8th term. … [2] (iii) Find the nth term. … [1] (b) The nth term of another geometric sequence is 3 2n 1 # - . (i) Show that the first term of this sequence is 3. [1] (ii) Complete the first four terms of this sequence. 3 , … , … , … [1]

3 0607/63/O/N/23 © UCLES 2023 [Turn over (c) The nth term of a different geometric sequence is . 36 0 5n 1 # - . Work out the first four terms of this sequence. … , … , …, … [2] (d) These are the first four terms of another sequence. 5 15 45 135 Find the nth term. … [2]

4 0607/63/O/N/23 © UCLES 2023 2 The first two terms of a geometric sequence are 16 and 12. Show that the 3rd term is 9. [2]

5 0607/63/O/N/23 © UCLES 2023 [Turn over 3 (a) A geometric sequence has: 2nd term = 12.5 3rd term = 62.5 . (i) Work out the 1st term. … [3] (ii) Find the nth term. … [1] (b) Another geometric sequence has: 2nd term = 24 5th term = 81 . Work out the 6th term. … [4]

6 0607/63/O/N/23 © UCLES 2023 4 These are the first three terms of a geometric sequence. x x 2 3 + x 11 6 - (a) (i) Give a reason why x x x x 2 3 2 3 11 6 + = + - . … … [1] (ii) Show that x x 7 18 9 0 2 - - = . [3] (b) Solve x x 7 18 9 0 2 - - = by factorisation. Give your answer as an integer and a fraction. x = … x = … [2]

7 0607/63/O/N/23 © UCLES 2023 [Turn over (c) There are two different geometric sequences where the first three terms are x, , x x 2 3 11 6 + - . Use your answer to part (b) to find the fourth term of each of these sequences. … … [4]

8 0607/63/O/N/23 © UCLES 2023 B MODELLING (QUESTIONS 5 TO 7) FILLING CONTAINERS (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at models for filling containers with water. In this task all containers are empty when the water starts to flow into them. 5 25 cm 20 cm 20 cm NOT TO SCALE A container is a cuboid 20 cm long, 20 cm wide and 25 cm high. (a) Calculate the volume of the container. … [2] (b) Water flows into the container at a constant rate of cm 50 3 per second. (i) Show that it takes 8 seconds to fill the container to a height of 1 cm. [2] (ii) How many seconds does it take to completely fill the container? … [1]

9 0607/63/O/N/23 © UCLES 2023 [Turn over (c) (i) Explain why the model for the time to fill the container is t = 8h where t seconds is the time taken and h cm is the height of water. … … [1] (ii) Work out the height of the water in the container after 1 minute. … [2] (iii) Sketch the model for the time to fill the container. t h 0 0 25 [2]

10 0607/63/O/N/23 © UCLES 2023 6 Volume, V, of a cone of radius r, height h. V r h 3 1 2 r = 24 cm NOT TO SCALE 16 cm A container is in the shape of a cone. The circular top has a radius of 16 cm and the container has a height of 24 cm. (a) Calculate the volume of the container. Give your answer correct to 3 significant figures. … [2]

11 0607/63/O/N/23 © UCLES 2023 [Turn over (b) 3 cm NOT TO SCALE x Water is poured into the container to a height of 3 cm. The surface of the water is a circle with radius x cm. (i) Show that x = 2. [2] (ii) Work out the volume of water in the container. … [2] (iii) Water flows into the empty container at a constant rate of cm 20 3 per second. Work out the time it takes to fill the container to a height of 3 cm. … [2]

12 0607/63/O/N/23 © UCLES 2023 (c) h cm NOT TO SCALE r The container is emptied. Water is now poured into the container to a height of h cm. The surface of the water is a circle with radius r cm. (i) Water flows into the container at a constant rate of cm 20 3 per second. The time taken to fill the container to a height of h cm is t seconds. Show that the model for the time to fill the container is t h 135 3 r = . [3] (ii) How many seconds does it take to completely fill the container? … [2]

13 0607/63/O/N/23 © UCLES 2023 [Turn over (iii) Sketch the model for the time to fill the container. t h 0 0 24 [2] (d) The container is emptied. Water is now poured into the container at a different rate. The model for the time to fill the container is now t h 189 3 r = . Find the rate of flow for this model. … [2]

14 0607/63/O/N/23 © UCLES 2023 7 The containers in Question 5 and Question 6 are both empty. Water flows into the container in Question 5 at a rate of cm 50 3 per second. At the same time water flows into the container in Question 6 at a rate of cm 0 2 3 per second. (a) Find the height of water when it is the same in both containers. … [2] (b) Find the time it takes for the water to reach this height. … [1]

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16 0607/63/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 7 printed pages. © UCLES 2023 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 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.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 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.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 3 of 7 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

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 4 of 7 Question Answer Marks Partial Marks 1(a)(i) 1024 1 1(a)(ii) 48 or 256 × 44 oe or 1024 × 43oe or 1024, 4096, 16384 C1 65536 1 1(a)(iii) 4n oe 1 1(b)(i) 3 × 20 1 1(b)(ii) 6, 12, 24 1 1(c) 36, 18, 9, 4.5 2 B1 for 3 terms correct 1(d) 5 × 3n–1 2 B1 for [5 ×] 3k for some k where k is an expression in terms of n 2 12 ×12 16 oe or 16 × 3 1 12 16 −       oe leading to 9 2 B1 for 12 16 oe or 16 12 oe seen 3(a)(i) 62.5 ÷ 12.5 C1 12.5 ÷ their (62.5 ÷ 12.5) C1 2.5 oe 1 3(a)(ii) their 2.5 × (their (62.5 ÷ 12.5))n – 1 1 FT 3(b) 24k3 = 81 oe with one further correct step leading to 1.5 oe or 3 81 24 = 1.5 C2 C1 for 24k3 = 81 or 81 24 oe or 1.5 used as multiplier 81 × their 1 3 81 24       oe C1 121.5 oe 1 4(a)(i) Correct comment referring to • Multiplier/divisor/scale factor/common ratio and • Equal to the same 1 4(a)(ii) Eliminating fractions e.g. 2 (2 + 3) = (11 6) x x x − oe 1 2 2 (2 + 3) = 4 + 6 + 9 x x x+6x 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 5 of 7 Question Answer Marks Partial Marks Rearrangement with one further correct step leading to 7x2 ̶ 18x – 9 = 0 . 1 4(b) (7x + 3)(x – 3 ) = 0 1 x = 3 and x = ̶ 3 7 1 If 0 scored SC1 for (7x –3)(x +3) and x = 3 7 and x = ̶ 3 4(c) Two correct sequences of three terms or nth term 3, 9, 27 or 3 × 3n–1 and ̶ 3 7 , 15 7 , 75 7 − C2 FT their 3 and their ̶ 3 7 but not if decimal C1FT for one correct sequence of three terms 81 375 7 or 4 53 7 2 B1 for each 5(a) 20 × 20 × 25 C1 10 000 cm3 1 5(b)(i) Full correct method shown. e.g. 20 × 20 [× 1] ÷ 50 = 8 or 10 000 ÷ 25 ÷ 50 = 8 2 B1 for 400 [× 1] ÷ 50 = 8 or B1FT their 10 000 ÷ 25 or their 10 000 ÷ 50 or their 10 000 ÷ 1250 5(b)(ii) 200 1 5(c)(i) Every 8 seconds the height goes up 1 cm oe 1 5(c)(ii) Substituting t = 60 into8 = h t (implied by 60 ÷ 8) C1 7.5 cm oe 1 5(c)(iii) Correct sketch 1 Ruled straight line from the origin with positive gradient reaching approx. h = 25 Labelling t axis and line goes through approx. (25, 200) or (25, 200) marked on the line above h = 25 C1 6(a) π×16×16×24÷3 oe C1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 6 of 7 Question Answer Marks Partial Marks 6430 cm3 1 6(b)(i) [x =] 16 3 24  oe or [x =] 3 16 24  oe or [x =] 16 ÷ 24 3 oe leading to [x =] 2 2 B1 for 16 3 = or = 3 24 16 24 x x or scale factor = 3 24 or 24 3 oe 6(b)(ii) 2 1 ×π×2 ×3 3 C1 12.6 cm3 1 6(b)(iii) their 12.6 ÷ 20 C1 FT 0.628 seconds 1 6(c)(i) t = 2 3 1 2 π 3 3 20 h       leading to 3 π 135 h 3 B2 for 2 1 2 π 3 3 h h       oe or B1 for 16 = 24 r h oe 6(c)(ii) 3 π24 135 or their 6(a) ÷ 20 C1 322 1 6(c)(iii) Correct Sketch 1 Constantly increasing curve from the origin reaching approx. h = 24 Labelling t axis and curve goes through approx. (24, 322) or (24, 322) marked on the line above h = 24 C1 0 322 0 24

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2023 © UCLES 2023 Page 7 of 7 Question Answer Marks Partial Marks 6(d) 2 3 1 2 π 3 3 Rate h       = 3 π 189 h oe or  189 20 135 = oe or 3 π24 189 or cm3 per second oe C1 28 1 7(a) 3 π 135 h = 8h or a sketch of both graphs. C1 18.5 cm 1 7(b) 148 s 1 If 0 or C1 only scored in 7(a) and 7(b) then also SC1 for reversed correct answers