Cambridge IGCSE Mathematics - International 0607 — 2022 May/June 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) 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 6) and part B (Questions 7 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 [ ]. DC (LK/CT) 303199/3 © UCLES 2022 * 7 2 8 9 1 5 8 1 6 8 *

2 0607/62/M/J/22 © UCLES 2022 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 6) OPPOSITE CORNERS (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation is about the difference between the products of the numbers in the opposite corners of a square window on a grid. To calculate the opposite difference for any window: • multiply the numbers in the opposite corners • subtract the smaller answer from the larger answer. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 Consecutive even numbers fill a grid of width 10 as shown. The grid continues downwards. A 2 by 2 window moves on the grid. Example This is the first window. 2 4 22 24 22 4 88 2 24 48 # # = = 88 48 40 - = The opposite difference is 40.

3 0607/62/M/J/22 © UCLES 2022 [Turn over 1 Use the grid to complete each window and find the opposite difference. 14 34 36 34 # … = … 14 # 36 = … … - … = … Opposite difference = … 66 68 86 … 150 152 … [3]

4 0607/62/M/J/22 © UCLES 2022 2 A 3 by 3 window moves on the same grid. (a) Complete the corner squares in the first window. 2 6 [1] (b) Complete the opposite difference calculations for this window. … # 6 = … 2 # … = … … - … = … [1] (c) Complete the corner squares for each window and find the opposite difference. 4 44 … 10 54 … 174 … [3]

5 0607/62/M/J/22 © UCLES 2022 [Turn over 3 (a) Copy the opposite differences that you have found and complete the table. Size of window Opposite difference 2 by 2 ( ) 2 1 2 - = 1 3 by 3 ( ) 1 3 2 - = 4 4 by 4 ( ) 1 4 2 - = 9 5 by 5 w by w 40( ) You may use this grid, which continues downwards, to help you. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 98 100 102 104 106 108 110 112 114 116 118 120 [4] (b) Find the greatest possible opposite difference for a window on this grid. … [2]

6 0607/62/M/J/22 © UCLES 2022 4 Another grid of consecutive even numbers has width 7 units. The diagram shows the start of the grid. 2 4 6 8 10 12 14 16 The diagram shows a 2 by 2 window on the grid. n is the first number in the window. n n + 2 (a) Complete the window using expressions in terms of n. [2] (b) Use part (a) to show that the opposite difference for a 2 by 2 window is 28. [2]

7 0607/62/M/J/22 © UCLES 2022 [Turn over 5 A 2 by 2 window moves on a grid of width g, with squares numbered 2, 4, 6, … . Use algebra to find an expression for the opposite difference on this grid. Give your answer in its simplest form. You may use this diagram to help you. … [4]

8 0607/62/M/J/22 © UCLES 2022 6 (a) A square window of side x 1 + moves on a grid of width g, with squares numbered 2, 4, 6, … . n is the first number in the window. n x + 1 x + 1 Show that an expression for the opposite difference using this window is gx 4 2, where x can be any positive integer. [4]

9 0607/62/M/J/22 © UCLES 2022 [Turn over (b) A square window moves on a grid numbered 2, 4, 6, … . The opposite difference is 144. Use your answer to part (a) to find all the ways this is possible. [4]

10 0607/62/M/J/22 © UCLES 2022 B MODELLING (QUESTIONS 7 TO 9) CRICKETS AND TEMPERATURE (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at the connection between the temperature and the number of times a cricket, a small insect, makes a chirping sound. Amos Dolbear (1837 to 1910) was an American scientist who suggested a relationship between the temperature and the number of times a Snowy Tree Cricket makes a chirp. He measured the air temperature and counted the number of chirps that a cricket makes in a given time. The table shows his data. Temperature, T °C Number of chirps in 60 seconds, N 14 83 16 88 17 102 19 106 21 129 22 138 23 157 25 180 27 208 7 (a) Calculate the mean temperature and the mean number of chirps. Temperature … Number of chirps … [2] (b) Plot the data on the grid on page 11. The first five points have been plotted for you.

11 0607/62/M/J/22 © UCLES 2022 [Turn over T N 7010 15 20 25 30 80 90 100 110 120 130 140 150 160 170 180 190 200 210 220 Number of chirps in 60 seconds Temperature, °C [2] (c) A model for the number of chirps, N, is the regression line for N in terms of T. Find this model. … [2] (d) Draw the graph of the model on the axes. [2] (e) A Snowy Tree Cricket chirps 170 times in 60 seconds. Use your model to find the temperature. … [2]

12 0607/62/M/J/22 © UCLES 2022 (f) Amos Dolbear originally counted the number of chirps in 13 seconds and measured the temperature in degrees Fahrenheit, F. To change the temperature from T °C to F, use . F T 1 8 32 = + . (i) Complete the table to change the data to the form Amos Dolbear used. All the data in the table is correct to the nearest integer. Temperature Number of chirps T °C F in 60 seconds in 13 seconds 14 57 83 18 16 61 88 19 17 63 102 22 19 106 21 129 22 138 23 73 157 34 25 77 180 39 27 81 208 45 [4] (ii) Amos Dolbear suggested a simple model to find the temperature, F, from the number of chirps. Add 40 to the number of chirps in 13 seconds to find the temperature, F. Is this a suitable model for the data? Give a reason for your answer. … … [1]

13 0607/62/M/J/22 © UCLES 2022 [Turn over 8 This is another model for the data on page 10. . N T aT b 0 5 2 = + + where a and b are constants (a) Use the information in the table on page 10 for temperatures of 16 °C and 27 °C to find the value of a and the value of b, each correct to the nearest integer. Write down the model. a = … b = … . N T 0 5 2 = … [5] (b) Sketch the graph of the model on the axes on page 11. [2] (c) Explain how suitable the model is for the data on page 10. … … [1]

14 0607/62/M/J/22 © UCLES 2022 9 The number of chirps per second, A, made by a type of African cricket is counted. These crickets do not chirp at higher or lower temperatures. The graph shows the results and a linear model. T A 1.517 17.5 18 18.5 19 19.5 20 20.5 21 21.5 22 2 2.5 3 3.5 4 Number of chirps per second Temperature, °C (a) (i) Use the points (17.5, 2.35) and (21, 3.5) to find a linear model for A in terms of T. … [4]

15 0607/62/M/J/22 © UCLES 2022 (ii) A is the number of chirps per second. N is the number of chirps per minute. Use your model for A to write a linear model for N. … [1] (b) Write two statements comparing the chirping of the Snowy Tree Cricket and this African cricket. Statement 1 … … … Statement 2 … … … [2]

16 0607/62/M/J/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 9 printed pages. © UCLES 2022 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 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.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 2 of 9 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 May/June 2022 © UCLES 2022 Page 3 of 9 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

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 4 of 9 Opposite Corners Question Answer Marks Partial Marks 1 16 40 88 40 170 172 40 2 B1 for two from 16, 88, 170 and 172 B1 for 40 seen three times One set of calculations for the opposite difference for the second or third grid C1 86×68 – 66×their88 or their170×152  150×their172 2(a) 42 46 1 2(b) their42 × 6 = their252 2 × their46 = their92 their 252 – their 92 1 2(c) 8 48 160 2 B1 for four from 8, 48, 14, 50, 134, 138 and 178 B1 for 160 seen three times 14 50 160 134 138 178 160 Complete set of calculations for a difference C1 44  their8 – 4  their48 or their50  their14 10  54 or 174  their138  their134  their178 3(a) Size of window Working Difference 2 by 2 (2 – 1)2 = 1 40 3 by 3 (3 – 1)2 = 4 160 4 by 4 (4 – 1)2 = 9 360 5 by 5 (5 – 1)2 = 16 640 w by w (w – 1)2 isw 40(w – 1)2 isw 3 B1 for each row or B1 for the 3 cells in the working columns A correct calculation for a 4 by 4 or 5 by 5 window C1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 5 of 9 Question Answer Marks Partial Marks 3(b) 40 (10 – 1)2 or 40 × 92 or correct calculation from grid C1 FT substitution of 10 in their 40(w – 1)2 3240 1 4(a) n + 14 n + 16 2 B1 for each 4(b) (n + 14)×(n + 2) = n2 + 16n + 28 M1 FT their (n + 14) if binomial n2 + 16n + 28 – n2  16n = 28 M1 5 n n + 2 n + 2g n + 2g + 2 oe C2 C1 for each (n + 2g) (n + 2) = n2 + 2ng + 2n + 4g C1 FT their (n+ 2g) and their (n + 2) if at least binomials include n and g [–] n(n + 2g + 2) and – n2 – 2ng – 2n giving 4g final answer 1 5 Alternative method 2 4 2 + 2g 2 + 2g + 2 oe C2 C1 for correct expression in 2nd row 4(2 + 2g) = 8 + 8g C1 FT their even number in first cell and their expression in g – 2(4 + 2g) = – 8 – 4g giving 4g as final answer 1 FT their even number in first cell leading to 4g 6(a) n + 2x n + 2gx oe n+ 2gx + 2x oe 2 B1 for any two correct (n + 2x) (n + 2gx) = n2 + 2nx + 2ngx + 4gx2 1 FT their (n + 2x) and their(n + 2gx) if at least binomials include n, g and x. [–] n(n + 2gx + 2x) and – n2 – 2ngx –2nx leading to 4gx2 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 6 of 9 Question Answer Marks Partial Marks 6(b) gx2 = 36 or 2 36  g x or 6  x g or for substituting for g or x in 4gx2 = 144 C1 2 by 2 window on a grid of width 36 3 by 3 window on a grid of width 9 4 by 4 window on a grid of width 4 3 B2 for x = 1 and g = 36 and x = 2 and g = 9 and x = 3 and g = 4 soi or B1 for one of the above soi 7(a) [Temperature] 20.4[4…] [Number of chirps] 132[.33…] 2 B1 for each 7(b) 4 points correctly plotted 2 B1 for 3 points correctly plotted 7(c) N = 9.72T – 66.4 2 B1 for 9.72T – 66.4 or N = 9.72T – k or N = k T – 66.4 or N = 9.7T – 66 7(d) Correct ruled straight line through their mean point 2 B1 for ruled straight line through their mean point with positive gradient or B1 for ruled straight line, not through their mean point, within tolerance. 7(e) Line N = 170 drawn or substitution N = 170 in their model C1 24 to 25 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 7 of 9 Question Answer Marks Partial Marks 7(f)(i) One calculation to convert temperature e.g. 1.8  19 + 32 and correct calculation for finding N e.g. 106  13 60 oe C2 C1 for a calculation to convert temperature e.g. 1.8  19 + 32 or correct calculation for finding N e.g. 106 oe Temperature C to F 19 66 21 70 22 72 Number of chirps 106 23 129 28 138 30 2 B1 for three correct values 7(f)(ii) Yes: The temperatures are around 40 more oe 1 Expect good fit oe as reason

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 8 of 9 Question Answer Marks Partial Marks 8(a) 88 = 0.5  162 + 16a +b C1 208 = 0.5  272 + 27a +b C1 Correct isolation of variable(s) and correct subtraction or correct substitution and 11a = –116.5 or –11a = 116.5 or 22a = –233 or –22a = 233 OR Correct isolation of b in both equations and sketch of intersecting straight lines C2 FT their equations if the same form. C1 for one of the first two. OR C1 for either [a =] 11 [b =] 129 B1 8(b) Correct sketch 2 B1 for correct shape B1 dependent for passing through (16, 88) and (27, 208) within tolerance 8(c) Good fit oe 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 9 of 9 Question Answer Marks Partial Marks 9(a)(i) [gradient or m] 3.5 2.35 21 17.5   or 2.35 = 17.5m + c and 3.5 = 21m + c and 0.329 or 0.3285 to 0.3286 or 0.33 or 23 70 or 1.15 3.5 or 1.15 = 3.5m oe C2 C1 for [gradient or m] 3.5 2.35 21 17.5   or 2.35 = 17.5m + c and 3.5 = 21m + c (17.5, 2.35) or (21, 3.5) correctly substituted in A = (their gradient)T + c C1 FT their gradient A = 0.329T  3.4 B1 9(a)(ii) N = 19.7T – 204 or N = 60 (0.329T – 3.4) oe isw 1 FT their model in (a)(i) 9(b) Two correct statements from: African cricket has a smaller range of temperatures oe African cricket starts chirping at a higher temperature oe African cricket stops chirping at a higher temperature oe African cricket has a smaller range of chirps/second oe African cricket has greater number of chirps per minute oe African cricket has greater rate of increase of chirps oe The increase in the rate of increase of chirps of the Snowy Tree cricket is more than the increase in the rate of increase of chirps of the African cricket. oe African cricket number of chirps are more varied oe 2 B1 for one statement