Cambridge IGCSE Mathematics - International 0607 — 2021 May/June Paper 5 · Variant 1

This document has 8 pages. Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 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 [ ]. DC (LK/SW) 207496/1 © UCLES 2021 [Turn over * 9 0 3 8 2 8 5 8 0 6 *

2 0607/51/M/J/21 © UCLES 2021 Answer all the questions. INVESTIGATION ROLLING SQUARE This investigation looks at the path of a point on a square as it rolls along the x-axis. A square of side 1 cm rolls along the x-axis. One roll is a turn of 90° clockwise about its bottom right corner. 0.5 1 1.5 0.5 1 1.5 Diagram 1 Starting position 2 2.5 3 y x 0 0 0.5 1 1.5 0.5 1 1.5 Diagram 2 After one roll 2 2.5 3 y x Position 1 Centre of rotation Position 1 Position 2 Centre of rotation Diagram 1 shows the square in Position 1. One side of the square is bold to help show the rotation. The centre of the square is (0.5, 0.5). Diagram 2 shows the square rolled 90° clockwise about (1, 0) to Position 2. 1 To get to Position 3 the square rolls 90° clockwise about (2, 0). To get to Position 4 the square then rolls 90° clockwise about (3, 0). (a) On the diagram below, draw the square in Position 4, Position 5 and Position 6. 0 1 2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y x [2] (b) Complete this table to show the x-coordinate of the centre of the square in each position. You may use the diagram to help you. Position (n) 1 2 3 4 5 6 n x-coordinate 0.5 1.5 2.5 [2]

3 0607/51/M/J/21 © UCLES 2021 [Turn over (c) Find the x-coordinate of the centre of the square in Position 92. … [2] (d) 0 1 2 1 2 3 4 5 6 7 y x (i) The square rolls from Position 1 to Position n. The centre has moved a distance equal to the circumference of 1 circle. The radius, r, of the circle is half the diagonal of the square. (a) Write down the number of rolls needed. … [1] (b) Write down the value of n. … [1] (ii) Circumference, C, of circle, radius r. C = 2rr y r x Hypotenuse, r, of a right-angled triangle with sides r, x and y. r x y 2 2 2 = + (a) Show that the radius of the circle is 0.707 cm, correct to 3 decimal places. [2] (b) Find the length of the arc that the centre of the square moves along from Position 1 to Position 2. … [2]

4 0607/51/M/J/21 © UCLES 2021 2 The side of the square is now 2 cm. 0 1 2 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y x The square rolls along the x-axis in the same way as in Question 1. (a) Complete the table of x-coordinates of the centre of the square in different positions. Position (n) 1 2 3 4 5 6 n x-coordinate 1 3 [3] (b) Find the coordinates of the centre of the square in Position 35. ( … , … ) [3] 3 (a) The side of the square is now 3 cm. Complete the table of x-coordinates of the centre of the square in different positions. You may use the diagram below to help you. Position (n) 1 2 3 4 5 6 n x-coordinate 1.5 0 1 2 3 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 y x [3]

5 0607/51/M/J/21 © UCLES 2021 [Turn over (b) The side of the square is now 4 cm. Complete the table of x-coordinates of the centre of the square in different positions. Position (n) 1 2 3 4 5 6 n x-coordinate 2 [2] 4 Write your expressions from Questions 1(b), 2(a) and 3 in the table below. Complete the table using any patterns you notice. Side of square (w cm) x-coordinate in Position n 1 2 3 4 5 w [4]

6 0607/51/M/J/21 © UCLES 2021 5 A square of side w cm rolls from Position 1 to Position 120. At Position 120, the x-coordinate of the centre of the square is 2151. Find the value of w. … [3]

7 0607/51/M/J/21 © UCLES 2021 [Turn over 6 A square of side a cm is in Position 1. The coordinates of the centre of the square are (11, k ). (a) Find the value of k and the value of a. k = … a = … [2] (b) Find the coordinates of the top right corner of the square. ( … , … ) [1] (c) Write down the y-coordinate of the centre of the square in Position 400. … [1] Question 7 is printed on the next page.

8 0607/51/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. 7 A square rolls along the x-axis. For the top left corner give a reason why total distance moved in 2 rolls = total distance moved in 3 rolls. You may use this grid. … … [2]

This document consists of 5 printed pages. © UCLES 2021 [Turn over Cambridge IGCSE™ CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/51 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.

0607/51 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.

0607/51 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

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 4 of 5 Question Answer Marks Partial Marks 1(a) Three correct squares in positions 4, 5 and 6 1 Centres marked on these three squares C1 1(b) 3.5 4.5 5.5 n – [0].5 oe 2 B1 for 3.5 4.5 and 5.5 B1 for n – 0.5 oe 1(c) Correct substitution of 92 in their expression or 92 – 0.5 C1 FT their expression 91.5 1 FT their expression If 0 scored SC1 for (91.5, 0.5) 1(d)(i)(a) 4 1 1(d)(i)(b) 5 1 1(d)(ii)(a) 12 + 12 or 0.52 + 0.52 1 0.7071... 1 1(d)(ii)(b) [C =] 2 0.707 4 ×π× C1 1.11 or 1.110 to 1.111 1 2(a) For at least 3 further squares on diagram or at least 3 further centres marked or at least 3 differences of 2 seen C1 5 7 9 11 2 1 − n oe 2 B1 for 5 7 9 11 B1 for 2 1 − n oe 2(b) Correct substitution of 35 in their 2n – 1 C1 FT their expression (69, 1) 2 B1 for each If 0 scored SC1 for (1, 69) 3(a) 4.5 7.5 10.5 13.5 16.5 3 1.5 − n oe 2 B1 for 4.5 7.5 10.5 13.5 16.5 B1 for 3 1.5 − n oe For at least 3 squares on diagram or at least 3 centres marked or at least 3 differences of 3 seen C1 3(b) 6 10 14 18 22 4 2 − n oe 2 B1 for 6 10 14 18 22 B1 for 4 2 − n oe

0607/51 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2021 © UCLES 2021 Page 5 of 5 Question Answer Marks Partial Marks 4 5 2.5 − n oe 1 2 −w wn oe 2 B1 for wn oe or 2 −w at least 3 differences of 1 or n oe in coefficient or at least 3 differences of −0.5 oe in constant C1 5 their 120 2   × −     w w = 2151 C1 FT their expression Correctly simplifies to a single term in w C1 FT their equation of the form wn + kw = 2151 18 nfww 1 6(a) [k =] 11 1 [a =] 22 1 6(b) 22, 22 1 FT their a 6(c) 11 1 FT their k 7 Squares in positions 1, 2 and 3 and correct arcs showing path of top left corner or 3 correct positions of top left corner C1 [on third roll] corner does not move oe 1