Cambridge IGCSE Mathematics - International 0607 — 2022 May/June Paper 5 · Variant 3

This document has 12 pages. Any blank pages are indicated. [Turn over DC (RW/SG) 219276/1 © UCLES 2022 * 3 5 9 5 5 7 0 3 6 1 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 Investigation (Core) May/June 2022 1 hour 10 minutes You must answer on the question paper. No additional materials are needed. INSTRUCTIONS ● Answer all questions. ● 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 36. ● The number of marks for each question or part question is shown in brackets [ ]. Cambridge IGCSE™

2 0607/53/M/J/22 © UCLES 2022 Answer all the questions. INVESTIGATION CIRCLES AND REGIONS This is an investigation into the number of regions formed by drawing lines on a circle. 1 Radii The diagrams show the number of regions inside a circle when 1 radius and 2 radii are drawn. The regions inside the circle are numbered. 1 1 radius 1 2 2 radii (a) (i) Draw radii on the circles below and number the regions. 5 radii 4 radii 3 radii [3] (ii) Complete the table. Number of radii Number of regions 1 1 2 2 3 4 5 6 [1] (b) Write a formula, in terms of n, for the number of regions, R, when there are n radii. … [1]

3 0607/53/M/J/22 © UCLES 2022 [Turn over 2 Diameters The diagrams show the number of regions inside a circle when 1 diameter and 2 diameters are drawn. 1 diameter 1 2 2 diameters 1 2 3 4 (a) Complete the table for 3, 4 and 5 diameters. You may use the empty circle to help you. Number of diameters Number of regions 1 2 2 4 3 4 5 [3] (b) Write a formula, in terms of n, for the number of regions, R, when there are n diameters. … [2]

4 0607/53/M/J/22 © UCLES 2022 3 Chords In this investigation: • each chord must cut every other chord • only two chords may intersect at any point. The diagrams show the number of regions inside a circle when 1 chord, 2 chords and 3 chords are drawn. 3 chords 2 chords 1 chord 1 1 2 3 4 4 7 5 6 3 1 2 2 (a) Count the number of regions in the circle when 4 chords are drawn. … [1]

5 0607/53/M/J/22 © UCLES 2022 [Turn over (b) Complete this table. You may use the empty circle to help you. Number of chords Number of regions 1 2 2 4 3 7 4 5 6 22 [2] (c) This is a formula for the number of regions, R, when there are n intersecting chords. R n bn 2 1 1 2 = + + Find the value of b. … [3]

6 0607/53/M/J/22 © UCLES 2022 4 Tangents A region can be inside or outside the circle when the lines are tangents. These two diagrams both show a circle with 2 tangents and the regions numbered. The maximum number of regions for a circle with 2 tangents is 6. 2 4 5 1 3 2 1 3 5 4 6 (a) Give a reason why the first diagram does not have the maximum number of regions with 2 tangents. … … [1] (b) (i) Use this diagram to find the maximum number of regions when there are 3 tangents. … [1]

7 0607/53/M/J/22 © UCLES 2022 [Turn over (ii) Draw a fourth tangent on the diagram below to find the maximum number of regions. … [2] (c) Use your answers to part (b) to complete the table. Number of tangents Maximum number of regions 1 3 2 6 3 4 5 [2] (d) This is a formula for the maximum number of regions, R, when there are n tangents. R n bn 2 1 1 2 = + + Find the value of b. … [3]

8 0607/53/M/J/22 © UCLES 2022 5 Secants A secant is a straight line that intersects a circle at two points and extends outside the circle. In this investigation: • each secant must cut every other secant • only 2 secants may intersect at any point • secants must not intersect on the circumference of the circle. secant The diagrams show the number of regions when 1 secant and 2 secants are drawn. 1 secant 2 4 3 1 2 secants 2 5 4 7 6 3 8 1 (a) Draw a third secant on the diagram below to find the number of regions when there are 3 secants. Complete the table. Number of secants Number of regions 1 4 2 8 3 4 19 5 26 [2]

9 0607/53/M/J/22 © UCLES 2022 [Turn over (b) This is a formula for the number of regions, R, when there are n secants. R n bn c 2 1 2 = + + Find the value of b and the value of c. b = … c = … [5]

10 0607/53/M/J/22 © UCLES 2022 6 There are two circles. The first circle has chords drawn on it. The second circle has tangents drawn on it. The number of chords on the first circle is the same as the number of tangents on the second circle. Each circle has the maximum number of regions. One circle has 60 more regions than the other. (a) Find the number of straight lines on each diagram. … [2] (b) Find the larger number of regions. … [2]

11 0607/53/M/J/22 © UCLES 2022 BLANK PAGE

12 0607/53/M/J/22 © UCLES 2022 BLANK PAGE 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.

This document consists of 5 printed pages. © UCLES 2022 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 2022 MARK SCHEME Maximum Mark: 36 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.

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 2 of 5 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.

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 3 of 5 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

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 4 of 5 Question Answer Marks Partial Marks 1(a)(i) 3 radii with 3 regions numbered 4 radii with 4 regions numbered 5 radii with 5 regions numbered 3 B1 for each If 0 scored, SC1 for all correct radii without regions numbered 1(a)(ii) [3, 4, 5] 6 in correct row 1 1(b) R = n 1 2(a) One circle with 3, 4 or 5 diameters drawn C1 6, 8, 10 in correct rows 2 B1 for one correct 2(b) R = C1 2n 1 3(a) 11 1 3(b) 3 differences seen or correct diagram C1 16 in row 5 of table 1 3(c) Equation in b with correct substitution for R and n 1 One correct step to simplify their equation C1 1 2 oe 1 4(a) (tangents do) not intersect oe or (tangents are) parallel 1 4(b)(i) 10 1 4(b)(ii) 4th tangent drawn that intersects the other 3 tangents 1 15 1 4(c) Correct 2 differences seen C1 FT their values in table 21 on row 5 1 4(d) Equation in b with correct substitution for R and n 1 One correct step to simplify their equation C1 1.5 oe 1 5(a) Secant drawn intersecting other 2 secants 1 13 (in row 3 of table) 1

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2022 © UCLES 2022 Page 5 of 5 Question Answer Marks Partial Marks 5(b) Two equations in b and c with correct substitution for R and n 2 B1 for one Correct method to eliminate either b or c C1 (b =) 2.5 (c =) 1 2 B1 for each 6(a) 4(d) – 3(c) = 60 oe or better or list of results showing differences C1 FT their 4(d) and their 3(c) 60 1 6(b) Correct substitution of 60 into 4(d) or 3(c) C1 FT their 60 in their 4(d) or their 3(c) 1891 1