Cambridge IGCSE Mathematics - International 0607 — 2022 Feb/March Paper 6 · Variant 2

This document has 20 pages. Any blank pages are indicated. [Turn over Cambridge IGCSE™ DC (LK/JG) 301931/3 © UCLES 2022 * 5 9 8 2 6 2 2 9 8 1 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 Investigation and Modelling (Extended) February/March 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 3) and part B (Questions 4 to 8). ● 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 0607/62/F/M/22 © UCLES 2022 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 TO 3) SEQUENCES OF CENTRED POLYGONS (30 marks) You are advised to spend no more than 50 minutes on this part. This investigation looks at sequences of centred polygons. The first pattern in each sequence is a single dot. The sequence continues by adding polygons of increasing size around the central dot. 1 (a) Oliver draws this sequence of dot patterns called centred squares. Pattern 3 Pattern 2 Pattern 1 (i) Pattern 3 is drawn on the grid. Complete the diagram to show Pattern 4. [1]

3 0607/62/F/M/22 © UCLES 2022 [Turn over (ii) Complete the table. Pattern number, n 1 2 3 4 5 Number of dots 1 5 13 [2] (iii) Work out the number of dots in Pattern 7. … [2]

4 0607/62/F/M/22 © UCLES 2022 (b) Oliver draws the dot patterns of centred squares using dots and crosses. Pattern 3 Pattern 4 Pattern 2 Pattern 1 (i) Pattern 4 is drawn on the grid. Complete the diagram to show Pattern 5. [1] (ii) Complete the table. Pattern number, n Number of dots Number of crosses Total number of dots and crosses 1 1 0 1 2 1 4 5 3 9 4 13 4 16 5 6 [2]

5 0607/62/F/M/22 © UCLES 2022 [Turn over (iii) Complete the table. Pattern number, n Number of dots Number of crosses Total number of dots and crosses 1 1 12 = 0 02 = 1 0 1 2 2 + = 2 1 12 = 4 22 = 2 1 5 2 2 + = 3 9 32 = 4 22 = 3 2 13 2 2 + = 4 16 = 5 6 [2] (iv) Complete the formula for the total number of dots and crosses, T, in Pattern n. T = … [2] (v) Show that the formula for T is n n 2 2 1 2 - + . [1] (vi) Work out the total number of dots and crosses in Pattern 15. … [2]

6 0607/62/F/M/22 © UCLES 2022 2 This is the sequence of centred triangles. Pattern 3 Pattern 2 Pattern 1 (a) Complete the table. Pattern number, n 1 2 3 4 5 Number of dots, T 1 4 10 [3] (b) The formula for the number of dots, T, in Pattern n is T an bn c 2 = + + . (i) Give a reason why a is 2 3. … [1] (ii) Find the value of b and the value of c. b = … c = … [4]

7 0607/62/F/M/22 © UCLES 2022 [Turn over (c) There are 571 dots in Pattern k. Find the value of k. k = … [4]

8 0607/62/F/M/22 © UCLES 2022 3 The formula for the number of dots, T, in the sequence of centred hexagons is T n n 3 3 1 2 = - + . The number of dots in the nth centred hexagon is 6 more than the number of dots in the nth centred square. Find the value of n and the number of dots in the centred hexagon. n = … number of dots in the centred hexagon : … [3]

9 0607/62/F/M/22 © UCLES 2022 [Turn over B MODELLING (QUESTIONS 4 TO 8) DAYLIGHT HOURS (30 marks) You are advised to spend no more than 50 minutes on this part. This task looks at modelling the number of daylight hours throughout the year. The number of daylight hours is the length of time between sunrise and sunset. 4 Sophia looks at the number of daylight hours in London, England. She knows the shortest number of daylight hours, shortest day, is on 21 December. She knows the longest number of daylight hours, longest day, is on 21 June. She collects data for the sunrise and sunset times on the 21st day of every month. She works out the length of daylight in hours and minutes and converts it to a time in hours correct to 1 decimal place. Her results are in this table. Daylight hours in London Month Day Sunrise Sunset Daylight in hours and minutes Daylight in hours December 21 08 03 15 53 7 h 50 min 7.8 January 21 07 52 16 30 8 h 38 min 8.6 February 21 07 02 17 26 10 h 24 min 10.4 March 21 06 00 18 15 12 h 15 min 12.3 April 21 05 51 20 07 14 h 16 min 14.3 May 21 05 00 20 54 15 h 54 min 15.9 June 21 04 43 21 21 16 h 38 min 16.6 July 21 05 08 21 04 15 h 56 min 15.9 August 21 05 55 20 10 14 h 15 min 14.3 September 21 06 45 19 00 12 h 15 min 12.3 October 21 07 35 17 54 November 21 07 28 16 03 8 h 35 min 8.6 December 21 08 03 15 53 7 h 50 min 7.8 (a) Complete the row for October. [2]

10 0607/62/F/M/22 © UCLES 2022 (b) Sophia puts her results from the table on this graph. 12 10 8 6 4 2 0 2 4 6 8 10 0 1 3 5 7 9 11 13 17 15 14 16 18 1 3 5 7 9 11 12 Hours Months after 21 December Daylight hours in London (i) Plot the value for daylight hours on 21 October. [1]

11 0607/62/F/M/22 © UCLES 2022 [Turn over (ii) Write down the number of daylight hours on the shortest day and on the longest day in London. shortest day … longest day … [1] (iii) Show that the length of time halfway between the number of daylight hours on the shortest day and the number of daylight hours on the longest day is 12.2 hours. [1]

12 0607/62/F/M/22 © UCLES 2022 5 Sophia changes the order of her table. • She starts with 21 March. • She works out daylight hours above and below 12.2 . Daylight hours in London Month Day Sunrise Sunset Daylight in hours and minutes Daylight in hours, H Daylight in hours above 12.2 , h March 21 06 00 18 15 12 h 15 min 12.3 0.1 April 21 05 51 20 07 14 h 16 min 14.3 2.1 May 21 05 00 20 54 15 h 54 min 15.9 3.7 June 21 04 43 21 21 16 h 38 min 16.6 4.4 July 21 05 08 21 04 15 h 56 min 15.9 3.7 August 21 05 55 20 10 14 h 15 min 14.3 2.1 September 21 06 45 19 00 12 h 15 min 12.3 0.1 October 21 07 35 17 54 November 21 07 28 16 03 8 h 35 min 8.6 ̶ 3.6 December 21 08 03 15 53 7 h 50 min 7.8 ̶ 4.4 January 21 07 52 16 30 8 h 38 min 8.6 ̶ 3.6 February 21 07 02 17 26 10 h 24 min 10.4 ̶ 1.8 March 21 06 00 18 15 12 h 15 min 12.3 0.1 (a) Complete the row for October. [1]

13 0607/62/F/M/22 © UCLES 2022 [Turn over (b) Sophia puts her results from the table into this graph. 0 h x 9 8 10 11 12 7 6 5 4 3 2 1 1 2 3 4 5 – 5 Months after 21 March – 4 – 3 – 2 – 1 Number of hours above 12.2 0 Sophia models the number of daylight hours in London using the points on her graph. She decides on this model. . ( ) sin h x 4 4 30 = h is the number of daylight hours above 12.2 and x is the number of months after 21 March. Give a reason why Sophia uses (i) the number 4.4, … … [1] (ii) the number 30. … … [1]

14 0607/62/F/M/22 © UCLES 2022 (c) Sophia changes her model. H is the number of daylight hours and x is the number of months after 21 March. Complete the model for H. H = … [1] (d) (i) Use the model in part (c) to calculate the number of daylight hours on 21 April and 21 January. April … January … [4] (ii) Sophia thinks that the calculated values for April and January show that her model is a good one. Is she correct? Give a reason for your answer. … … [1] (iii) What assumption has Sophia made about the months for her model? … [1]

15 0607/62/F/M/22 © UCLES 2022 [Turn over 6 This table shows sunrise and sunset times for Tokyo in Japan on the 21st day of every month. Sunrise and sunset times in Tokyo Month Day Sunrise Sunset March 21 05 43 17 53 April 21 05 00 18 19 May 21 04 32 18 44 June 21 04 25 19 00 July 21 04 40 18 53 August 21 05 04 18 23 September 21 05 28 17 39 October 21 05 52 16 58 November 21 06 22 16 30 December 21 06 47 16 32 January 21 06 48 16 56 February 21 06 22 17 27 March 21 05 43 17 53 The shortest day is 21 December and the longest day is 21 June. Find a model for the number of daylight hours, H, in Tokyo. Write your model in the form ) ( sin H A x C B = + , where x is the number of months after 21 March and A, B and C are numbers to be found. H = … [6]

16 0607/62/F/M/22 © UCLES 2022 7 This is a model of the number of daylight hours in Cairo in Egypt. . ( ) sin H x 2 30 12 2 = + H is the number of daylight hours and x is the number of months after 21 March. (a) Use the model to estimate the number of daylight hours on the shortest day in Cairo. … [2] (b) What is the date of the shortest day in Cairo according to the model? … [1]

17 0607/62/F/M/22 © UCLES 2022 8 This is a model of the number of daylight hours in Melbourne in Australia. . ( ) . sin x H 2 4 30 180 12 2 = + + H is the number of daylight hours and x is the number of months after 21 March. (a) On the axes below sketch the models for the number of daylight hours: • in Cairo • in Melbourne. Months after 21 March H x 12 0 Daylight hours 15 9 [3] (b) Make a statement about the number of daylight hours in Melbourne on the shortest day in Cairo. … [1] (c) Find the dates when the number of daylight hours in Cairo is the same as it is in Melbourne. … [2]

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20 0607/62/F/M/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™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) February/March 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 February/March 2022 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 February/March 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. 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.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2022 © UCLES 2022 Page 3 of 7 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 February/March 2022 © UCLES 2022 Page 4 of 7 Question Answer Marks Partial Marks A INVESTIGATION SEQUENCES OF CENTRES POYGONS 1(a)(i) Correct diagram 1 1(a)(ii) Pattern number, n 1 2 3 4 5 Number of dots 1 5 13 25 41 2 B1 for 25 or their 25 +16 1(a)(iii) 61 seen or 41 + 20 + 24 or 16 20 24 C1 FT their 41 85 1 1(b)(i) Correct diagram 1 1(b)(ii) Pattern number, n Number of dots Number of crosses Total number of dots and crosses 4 9 16 25 5 25 16 41 6 25 36 61 2 B1 for two columns FT their dots and their crosses for the last column.

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2022 © UCLES 2022 Page 5 of 7 Question Answer Marks Partial Marks 1(b)(iii) Pattern Dots Crosses Total number 4 9 = 32 16 = 42 42 + 32 = 25 5 25 = 52 16 = 42 52 + 42 = 41 6 25 = 52 36 = 62 62 + 52 = 61 2 B1 for 5 cells correct or for the total column correct FT their values in (b)(ii) 1(b)(iv) n2 + (n – 1)2 isw 2 B1 for n2 or (n – 1)2 1(b)(v) [n2 + (n – 1)2] = n2 + n2 – n – n +1 = 2n2 – 2n + 1 oe 1 1(b)(vi) 2 × 152 – 2 × 15 + 1 or 152 + 142 or correct extension of sequence C1 421 1 2(a) For sketching Pattern 4 or for differences 3 6 9 12 C1 Pattern number, n 1 2 3 4 5 Number of dots 1 4 10 19 31 2 B1 for 19 B1 for 31 2(b)(i) One second difference of 3 seen and 2a = 3 1 2(b)(ii) Forming two equations in b and c. or 2 3 2 n   =     1.5, 6, 13.5, 24 C1 Correct method to find b or c. or remaining terms are –0.5, –2, –3.5, –5 C1 b = –1.5 c = 1 2 B1 for each 2(c) 1.5k2 – 1.5k + 1 = 571 C1 FT their b and c Attempt to solve their quadratic equation correctly C1 Sketch of their quadratic or factorisation attempt for their quadratic or correct formula Solution(s) marked on sketch or a correct factorisation or correct substitution into formula C1 FT their quadratic equation 20 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2022 © UCLES 2022 Page 6 of 7 Question Answer Marks Partial Marks 3 [n = ]3 [Number of dots =] 19 3 B2 for n = 3 or B1 for 3n2 – 3n + 1 = 2n2 – 2n +1 + 6 oe or for table of hexagon values B MODELLING DAYLIGHT HOURS 4(a) 10h 19min 10.3 or awrt 10.32 2 B1 for 10h 19min or correct decimal from their 10h 19min. 4(b)(i) Correct point (10, their 10.3) plotted 1 FT (10, their10.3) 4(b)(ii) (shortest day) 7.8 (longest day) 16.6 1 4(b)(iii) 7.8 16.6 2 + 1 5(a) their10.3 – 12.2 1 5(b)(i) Amplitude or its description = 4.4 or 16.6 7.8 2 − = 4.4 oe 1 5(b)(ii) [Period] 360 ÷ 12 = 30 oe 1 5(c) 4.4 sin 30 x + 12.2 1 5(d)(i) One of x = 1 [for April] x = 10 [for January] C1 4.4 sin30 + 12.2 or 4.4 sin300 + 12.2 or x = 1 or x = 10 shown on sketch of H = 4.4 sin 30 x + 12.2 C1 FT their H FT their x if stated [April] 14.4 B1 [January] 8.4 or 8.39 or 8.38[...] B1 5(d)(ii) No, because two values are insufficient to make that claim oe 1 FT their 5(d)(i) and condone Yes, because both are close to the actual values oe or No, because their values are very different oe 5(d)(iii) The months are all the same length oe 1

0607/62 Cambridge IGCSE – Mark Scheme PUBLISHED February/March 2022 © UCLES 2022 Page 7 of 7 Question Answer Marks Partial Marks 6 B = 30 soi 1 14 h 35 min and 14.6 [h] or 14.58[...] 9 h 45 min and 9.8 or 9.75 [h] C2 C1 for both h and min or both decimals correct in each month or for one month completely correct C = 12.2 soi 1 FT their decimal hours 14.6 – 12.2 or 12.2 – 9.8 or 14.6 9.8 2 − C1 FT their decimal hours and their C. A = 2.4 H = 2.4 sin 30 x + 12.2 1 FT their values if calculation seen 7(a) Substituting x = 9 or sin(30x) = –1 or amplitude = 2 oe and 12.2 – 2 or sketch of graph with minimum marked C1 If 0 scored, SC1 for substituting 10 to get 10.5 10.2 1 7(b) 21 December 1 8(a) Graphs correctly labelled Cairo and Melbourne. C1 Correct sketch. 2 B1 for correct shape with intersections at x = 0, approx. 6, 12 and approx halfway between horizontal axis and 15 B1 for correct shape with amplitude (Melbourne) > amplitude (Cairo) 8(b) It is the longest day oe or There are 14.6 h of daylight oe 1 8(c) 21 March 21 September 2 B1 for each If 0 scored: SC1 for 0 and 6 or 6 and 12 seen or for March and September 0 1 2 3 4 5 6 7 8 9 10 11 12 9 10 11 12 13 14 15 x H