Cambridge IGCSE Mathematics - International 0607 — 2021 May/June Paper 5 · Variant 2

This document has 12 pages. Any blank pages are indicated. DC (RW/FC) 207546/2 © UCLES 2021 [Turn over * 6 5 5 3 2 2 8 5 6 3 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 Paper 5 Investigation (Core) May/June 2021 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 0607/52/M/J/21 © UCLES 2021 Answer all the questions. INVESTIGATION NEAREST NEIGHBOURS This investigation looks at the distances from the origin (0, 0) to the centres of squares and hexagons that form a pattern. The diagram shows a pattern of congruent squares. A dot marks the centre of each square. The coordinates of some of the centres are (0, 0), ( 3 - , 2 - ) and (3, 1). y x 1 2 3 4 1 2 3 –3 –2 –1 –4 –3 –2 –1 0

3 0607/52/M/J/21 © UCLES 2021 [Turn over 1 (a) Complete this table, using a tick in each row, to show whether a point is • at the centre of a square • inside a square, but not at its centre • on the side of two squares • where four squares meet. The first three have been done for you. Point At the centre of a square Inside a square, not at its centre On the side of two squares Where four squares meet (0, 0)  ( 1 - , 0.5)  (0.5, 0.5)  (1.5, 1.5) (0, 1.5) (0.25, 0) (100.5, 99.5) [4] (b) Give a reason for your answer for (100.5, 99.5). … … [1]

4 0607/52/M/J/21 © UCLES 2021 2 Each dot on the grid marks the centre of a square. y x 1 2 3 1 2 3 –3 –2 –1 –3 –2 –1 0 The nearest dots to (0, 0) are ( 1 - , 0), (0, 1), (1, 0) and (0, 1 - ). These four dots are the 1st nearest neighbours. The 2nd nearest neighbours to (0, 0) are (1, 1), (1, 1 - ), ( 1 - , 1 - ) and ( 1 - , 1). All nearest neighbours have integer coordinates.

5 0607/52/M/J/21 © UCLES 2021 [Turn over (a) One of the 3rd nearest neighbours to (0, 0) is (2, 0). Find the other 3rd nearest neighbours and write down their coordinates. … [3] (b) Find the coordinates of all the 4th nearest neighbours to (0, 0). … … [3]

6 0607/52/M/J/21 © UCLES 2021 3 You can find the distance, d, from (0, 0) to the point (a, b) using Pythagoras’ Theorem. d a b 2 2 2 = + (a) Show that the distance of a 4th nearest neighbour from (0, 0) is 5. [2] (b) Here are four points and their coordinates. A(20, 5) B(7, 24) C( 7 - , 24) D(0, 25) Which of these points are a distance of 25 units from (0, 0)? … [3]

7 0607/52/M/J/21 © UCLES 2021 [Turn over (c) There are more than 10 nearest neighbours to (0, 0) with d 5 = . Four of them are (0, 5), (5, 0), ( 5 - , 0) and (0, 5 - ). On the grid, mark with a cross all the nearest neighbours to (0, 0) with d 5 = . y x 1 2 3 4 5 1 2 3 4 5 –3 –4 –5 –2 –1 –3 –4 –5 –2 –1 0 [4]

8 0607/52/M/J/21 © UCLES 2021 4 Here is a pattern of congruent regular hexagons. Each dot marks the centre of a hexagon. The triangle shown is an equilateral triangle. (a) (i) Explain why the triangle shown is an equilateral triangle. … [1] The dots at the centre of the hexagons have coordinates as shown on the grid below. y x 0 1 2 3 4 1 2 3 4 5 6 7 (7, 0) (5, 4) (ii) A line is made by joining the points (3, 0) and (4, 1). Work out the size of the acute angle between this line and the x‑axis. … [2]

9 0607/52/M/J/21 © UCLES 2021 [Turn over (b) The point at the centre of each hexagon has six 1st nearest neighbours. (i) Complete this list of the 1st nearest neighbours to the point (1, 1). (0, 1), (0, 2), (1, 2), … [2] (ii) Find, in terms of a and b, the coordinates of the six 1st nearest neighbours to the point (a, b). … … [3]

10 0607/52/M/J/21 © UCLES 2021 (c) A student suggests that an estimate of d, the distance from (0, 0) to the point (a, b), is d a b ab 2 2 = + + . The distance between any point and its 1st nearest neighbours on this grid is 1 cm. y x 0 2 3 4 –1 –3 – 4 – 5 –2 –1 –2 –3 –4 1 2 3 4 5 1 Does the student’s formula give a good estimate for the distance from (0, 0) to (4, 3 - )? Show how you decide. [5]

11 0607/52/M/J/21 © UCLES 2021 5 The circles show the 1st, 2nd, 3rd, and 4th nearest neighbours to the point X. X (a) Complete this table. Nearest neighbour 1st 2nd 3rd 4th 5th 6th 7th 8th Number of nearest neighbours 6 12 6 [2] (b) A computer calculates that there is a total of 9 points for a certain nearest neighbour distance. Explain why the computer is probably wrong. … [1]

12 0607/52/M/J/21 © UCLES 2021 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 the Cambridge Assessment Group. Cambridge Assessment is the brand name of the University of Cambridge Local Examinations Syndicate (UCLES), which itself is a department of the University of Cambridge. BLANK PAGE

This document consists of 5 printed pages. © UCLES 2021 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/52 Paper 5 (Core) May/June 2021 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 2021 series for most Cambridge IGCSE™, Cambridge International A and AS Level components and some Cambridge O Level components.

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 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.

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 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

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 4 of 5 Question Answer Marks Partial Marks 1(a) 4 B1 for each correct row. 1(b) Both coordinates have a 0.5 oe 1 Dependent on last row correct. 2(a) (0, 2), (–2, 0), (0, –2), [(2, 0)] 2 B1 for 2 correct Indication of at least 3 correct points C1 2(b) (1, 2), (1, –2), (–1, 2), (–1, –2) (2, 1), (2, –1), (–2, 1), (–2, –1) only 2 B1 for at least 4 correct with at most 2 wrong, ignoring repeats At least 4 points correctly indicated on grid C1 3(a) [d2 =] 12 + 22 oe 1 [d2 =] 5 leading to [d =] 5 1 Dependent on first mark 3(b) Correct use of Pythagoras for B, C or D C1 B, C and D 2 B1 for 2 correct or for answer of A, B, C and D 3(c) Attempt at circle centre (0, 0) and radius 5 or at least 4 lines drawn from centre to points C1 12 points plotted correctly 3 B2 for 8, 9, 10 or 11 correctly plotted points B1 for 4, 5, 6 or 7 correctly plotted points 4(a)(i) All angles in a triangle are 60° or all angles are the same or 180 ÷ 3 = 60 or 360 ÷ 6 = 60 or all sides/lines [of the triangle are] equal or distances between centres of hexagons [are]equal 1 4(a)(ii) 30° 1 Correctly drawn straight line or 60 ÷ 2 C1

0607/52 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 5 of 5 Question Answer Marks Partial Marks 4(b)(i) (1, 0) (2, 0) (2, 1) only 2 B1 for 3 correct with up to 3 extras or B1 for 2 correct with up to 2 extras Ignore repeats 4(b)(ii) (a, b + 1) (a + 1, b – 1) (a – 1, b) (a, b – 1) (a + 1, b) (a – 1, b + 1) only 3 B2 for at least 3 correct with at most 3 extras B1 for 2 correct with at most 2 extras 4(c) 2 2 3 4 3 4 + + −× or better 1 3.6[05…] or 13 1 Measured is 3.5 to 3.7 oe 1 Correct statement for their two values seen 1 FT their values cm stated for both quantities C1 FT their values Alternative 2 2 3 4 3 4 + + −× or better (1) Correct estimation method using right-angled triangle (1) Estimated is approx. 3.5 oe (1) Correct statement for their two values seen (1) FT their values cm stated for both quantities (C1) FT their values 5(a) Next circle or points indicated. C1 1 5(b) [Should probably] be [either] 6 s or 12 s oe 1