Cambridge IGCSE Mathematics - International 0607 — 2022 Oct/Nov Paper 6 · Variant 1

This document has 16 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ * 2 8 1 3 3 9 8 9 7 6 * DC (CJ/SG) 303203/2 © UCLES 2022 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 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 5) and part B (Questions 6 to 9). ● 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 0607/61/O/N/22 © UCLES 2022 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 5) ISOSCELES TRAPEZIUMS (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at equilateral triangles in isosceles trapeziums drawn on 1 cm isometric grids. There is a spare isometric grid on page 9. A unit triangle is an equilateral triangle of side length 1. In this task, all shapes can be divided into unit triangles. 1 There are 16 unit triangles in the equilateral triangle with side length 4. These are the first three equilateral triangles in a sequence. 1 2 3

3 0607/61/O/N/22 © UCLES 2022 [Turn over (a) Complete the table. Side length (x) 1 2 3 4 5 Number of unit triangles 1 16 [2] (b) Write down an expression, in terms of x, for the number of unit triangles. … [1] 2 These are the first three isosceles trapeziums in a sequence. The length of each sloping side is y. y = 1 y = 2 y = 3 (a) Complete the table. Length of sloping side (y) 1 2 3 4 Number of unit triangles 3 24 [1] (b) Find an expression, in terms of y, for the number of unit triangles. Give your answer in its simplest form. … [3]

4 0607/61/O/N/22 © UCLES 2022 3 These are the first three trapeziums in another sequence. The length of each sloping side is 1. The length of the shorter parallel side is w. w = 1 w = 2 w = 3 (a) Complete the table. Length of shorter parallel side (w) Number of unit triangles 1 22 - = 3 2 - 22 = 3 - = 7 4 - = [2] (b) Find an expression, in terms of w, for the number of unit triangles. Give your answer in its simplest form. … [2]

5 0607/61/O/N/22 © UCLES 2022 [Turn over 4 These are the first three trapeziums in another sequence. The length of each sloping side is 3. The length of the shorter parallel side is w. w = 1 w = 2 w = 3 (a) Complete the table. Length of shorter parallel side (w) Number of unit triangles 1 42 - = 15 2 - 22 = 3 - = 27 4 - = [2] (b) Find an expression, in terms of w, for the number of unit triangles. Give your answer in its simplest form. … [2] (c) Find the value of w when there are 93 unit triangles. … [2]

6 0607/61/O/N/22 © UCLES 2022 5 w y In each trapezium: • the length of the shorter parallel side is w • the length of each sloping side is y. (a) Show that an expression for the number of unit triangles is ( ) y y w 2 + . [3]

7 0607/61/O/N/22 © UCLES 2022 [Turn over (b) In a trapezium the length of the shorter parallel side is equal to the length of the sloping side. Is it possible for the number of unit triangles to be 300? [4]

8 0607/61/O/N/22 © UCLES 2022 (c) Find the lengths of the sides for all the trapeziums with exactly 160 unit triangles. [6]

9 0607/61/O/N/22 © UCLES 2022 [Turn over SPARE GRID

10 0607/61/O/N/22 © UCLES 2022 B MODELLING (QUESTIONS 6 TO 9) PIZZA BUSINESS (30 marks) You are advised to spend no more than 50 minutes on this part. This task is about the costs in a pizza business. Pablo is starting a pizza business. He collects information from pizza businesses to set up mathematical models. He assumes that he sells all the pizzas he makes. 6 Model 1 Pablo sets the following conditions: • He sells one type of pizza with a selling price of $10. • He sells n pizzas per week. • It costs him $2 to make one pizza. • His running costs are $400 per week. • He employs 2 people each costing $300 per week. This is how he works out his profit. profit = income from selling pizzas - his costs (a) Show that a model for the weekly profit, $P, in terms of n, is . P n 8 1000 = - [1] (b) Pablo wants a profit of at least $500 per week. Use the model to find the minimum number of pizzas he must sell per week. … [3]

11 0607/61/O/N/22 © UCLES 2022 [Turn over (c) Pablo expects sales to vary. He sets the following conditions. The pizza business: • opens 4 days per week • opens 6 hours per day • makes 16 pizzas per hour for half of the time • makes 8 pizzas per hour for the rest of the time. Can Pablo make a weekly profit of $500? Use the conditions above and your answer to part (b) to show how you decide. [3]

12 0607/61/O/N/22 © UCLES 2022 7 Pablo makes changes to the conditions. These are the conditions now: • He sells one type of pizza with a selling price of $x. • He sells n pizzas per week. • It costs him $2 to make one pizza. • His running costs of $400 increase by 50%. • He employs 3 people each costing $300 per week. (a) Use these conditions to show that the model for the weekly profit, $P, is ( ) P x n 2 1500 = - - . [3] (b) The pizza business is now open for 6 hours per day and 6 days per week. It continues to make 16 pizzas per hour for half of the time and 8 pizzas per hour for the rest of the time. Work out the number of pizzas the business can make per week. … [2]

13 0607/61/O/N/22 © UCLES 2022 [Turn over (c) Pablo wants a weekly profit of at least $500. Use the model in part (a) and your answer to part (b) to work out the minimum selling price of a pizza. … [3]

14 0607/61/O/N/22 © UCLES 2022 8 Model 2 Pablo makes a model for the relationship between the price of a pizza and the number of pizzas he sells. He uses the following information: • He sells 480 pizzas per week when the selling price is $5 per pizza. • He sells 60 pizzas per week when the selling price is $20 per pizza. Pablo assumes this relationship between the selling price of a pizza, $x, and the number of pizzas he sells per week, n. n ax b = + Write down and solve a pair of simultaneous equations to show that a 28 =- and b 620 = . [4] 9 Model 3 Pablo wants a model for the weekly profit, $P, in terms of the selling price, $x, of the pizzas. (a) Use Question 7(a) and Question 8 to show that a model for the weekly profit, in terms of x, is . P x x 28 676 2740 2 =- + - [3]

15 0607/61/O/N/22 © UCLES 2022 (b) Sketch the graph of the model on the axes. P x 0 25 [3] (c) Find the maximum weekly profit and the selling price that gives this. Maximum weekly profit … Selling price … [2] (d) Write an inequality for the values of x that give a weekly profit, $P, of at least $500. … [3]

16 0607/61/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 7 printed pages. © UCLES 2022 [Turn over Cambridge IGCSE™ MATHEMATICS 0607/61 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.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 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.

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 3 of 7 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

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 4 of 7 Question Answer Marks Partial Marks 1(a) 4, 9 and 25 2 B1 for 2 values correct 1(b) x2 1 2(a) 8, 15 1 2(b) two common differences of 2 seen or three of 1  3, 2  4, 3  5, 4  6 seen or 1, 4, 9, [16] or three of 4 – 1, 9 – 1, 16 – 1, 25 – 1 seen C1 next step shown in difference method or two from 1 + 2 = 3, 2 + 2 = 4, 3 + 2 = 5, 4 + 2 = 6 oe or 3, 8, 15, [24] subtract 1, 4, 9, [16] or (y + 1)2 ‒ 1 C1 y2 + 2y or y(y + 2) isw 1 3(a) 32 – 1[2] = 3 32 – 22 = 5 42 – 32 = 7 52 – 42 = 9 2 B1 for five, six or seven correct entries 3(b) (w + 1)2 ‒ w2 seen or three differences of 2 seen C1 2w + 1 isw 1 4(a) 42 – 1[2] = 15 52 – 22 = 21 62 – 32 = 27 72 – 42 = 33 2 B1 for five, six or seven correct entries 4(b) (w + 3)2 ‒ w2 seen or three differences of 6 seen C1 6w + 9 or 3(2w + 3) isw 1 4(c) 6w + 9 = 93 C1 FT their 4(b) = 93 14 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 5 of 7 Question Answer Marks Partial Marks 5(a) Diagram with right-hand side labelled y and base labelled w + y C1 (w + y)2 – w2 1 Expansion to w2 + 2yw + y2 – w2 leading to y (y + 2w) 1 Alternative 4w + 4 seen (C1) 2, 4, 6 expressed as 2y and 1, 4, 9 expressed as y2 (1) Factorisation of 2wy + y2 to y (y + 2w) (1) 5(b) y = w stated or trapezium labelled with one variable C1 y(y + 2y) = 300 oe or w(w + 2w) = 300 oe or (y + y)2 – y2 = 300 oe or (w+ w)2 – w2 = 300 oe or y(3y) = 300 or w(3w) = 300 or 3y2 = 300 or 3w2 = 300 C1 y2 = 100 or w2 = 100 seen 1 y = 10 or w = 10 and Yes oe 1 Alternative y = w stated or diagram labelled with one variable (C1) Correct substitution of a value into y(y + 2w) oe (C1) y = 10 or w = 10 and Yes oe (2) B1 for y = 10 or w = 10

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 6 of 7 Question Answer Marks Partial Marks 5(c) At least two from 1  160, 2  80, 4  40, 5  32, 8  20, 10  16 C1 Two correct substitutions for y in y(y + 2w) =160 oe or two correct substitutions for y and w in y(y + 2w) oe C1 y 2 4 8 10 w 39 18 6 3 w + y 41 22 14 13 4 B3 for three correct pairs of y and w or two correct triples B2 for two correct pairs of y and w or one correct triple or 3 correct values for y or w B1 for one correct pair of y and w or 2 correct values for y or w 6(a) 10n – 2n – 400 ‒ 2  300 oe 1 6(b) 8n ‒ 1000 ≥ 500 C1 188 2 B1 187.5 seen 6(c) 12  16 + 12  8 or 12(16 + 8) or 4(16  3 + 8  3) or 24  3  4 or C1 FT their 4 6 2        or 12  16 etc. 288 A1 Yes oe, because 288 > 188 oe 1 FT their 188 and their 288 7(a) 400  1.5 or 400  150 100 or 400 + 200 or 600 1 300  3 or 900 1 xn – 2n – 600 ‒ 900 1 7(b) 18  16 + 18  8 or 18(16 + 8) or 6(16  3 + 8  3) or 1.5  288 oe C1 FT their 18 FT their 288 from 6(c) 432 1 7(c) (x – 2)  432 – 1500 [= P or 500] C1 FT their 432 Correct step towards isolating x or x – 2 C1 FT their 432 [$]6.63 or 6.629[…] cao 1

0607/61 Cambridge IGCSE – Mark Scheme PUBLISHED October/November 2022 © UCLES 2022 Page 7 of 7 Question Answer Marks Partial Marks 8 480 = 5a + b 60 = 20a + b 2 B1 for each correct method to eliminate one variable 1 Substitution of their (a or b) leading to 620 or –28 or correct method to eliminate and find correct second variable 1 9(a) [P =] (x – 2)(‒28x + 620) ‒ 1500 1 Two of: ‒28x2, 56x, 620x, ‒1240 1 Correct completion to –28x2 + 676x – 2740 1 9(b) Correct sketch 1 x-axis intercepts at approx. 5 and 19 C1 Indication of scale on P-axis or maximum value C1 9(c) [$]1340.14 [$]12.07 2 B1 for each If 0 scored SC1 for correct values reversed. 9(d) Line on graph at P = 500 or correct use of formula to solve – 28x2 + 676x – 3240 C1 6.59 ⩽ x ⩽ 17.5 2 B1 for 6.59 ⩽ x or x ⩽ 17.5 If 0 scored SC1 for both values correct with incorrect or missing inequality signs