Cambridge IGCSE Mathematics - International 0607 — 2018 May/June Paper 5 · Variant 3

This document consists of 7 printed pages and 1 blank page. DC (LEG/SG) 153610/1 © UCLES 2018 [Turn over * 5 2 1 3 8 5 1 9 4 2 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 2018 1 hour Candidates answer on the Question Paper. Additional Materials: Graphics Calculator READ THESE INSTRUCTIONS FIRST Write your Centre number, candidate number and name on all the work you hand in. Write in dark blue or black pen. Do not use staples, paper clips, glue or correction fluid. You may use an HB pencil for any diagrams or graphs. DO NOT WRITE IN ANY BARCODES. Answer all the questions. You must show all relevant working to gain full marks for correct methods, including sketches. In this paper you will also be assessed on your ability to provide full reasons and communicate your mathematics clearly and precisely. At the end of the examination, fasten all your work securely together. The total number of marks for this paper is 24. Cambridge International Examinations Cambridge International General Certificate of Secondary Education

2 0607/53/M/J/18 © UCLES 2018 Answer all the questions. INVESTIGATION ESTIMATING r This investigation is about using relative frequency to estimate the value of r. Area of rectangle = length # width. Area, A, of circle radius r is A= rr2. Lee draws circles on rectangular pieces of paper. He drops grains of rice at random onto the pieces of paper. He counts the number of grains of rice inside each circle. 1 Lee draws a circle of radius 5 cm on a rectangular piece of paper measuring 40 cm by 20 cm. 5 cm 40 cm 20 cm NOT TO SCALE (a) (i) Find the area of the rectangle. … (ii) The probability, p, that a grain of rice lands inside the circle is p = area of circle area of rectangle. The area of the circle is 25 # r. Show that p is approximately 0.098 for this piece of paper. (b) Lee drops 10 grains of rice at random onto the piece of paper. The diagram shows the result. NOT TO SCALE Key: represents one grain of rice.

3 0607/53/M/J/18 © UCLES 2018 [Turn over (i) How many of the grains of rice are inside the circle? … (ii) The relative frequency that a grain of rice is inside the circle = number of grains of rice inside the circle total number of grains of rice dropped . Find the relative frequency that a grain of rice is inside the circle. … (c) Lee drops 10 more grains of rice at random onto the piece of paper. NOT TO SCALE Show that the relative frequency that a grain of rice is inside the circle is 0.15 . (d) The relative frequency that a grain of rice is inside the circle gives an estimate for the probability, p. The area of the circle is 25 # r. Use area of circle area of rectangle = 0.15 to show that an estimate for r is 4.8 .

4 0607/53/M/J/18 © UCLES 2018 2 Lee draws a circle of radius 10 cm on a rectangular piece of paper measuring 30 cm by 20 cm. NOT TO SCALE 10 cm 30 cm 20 cm (a) Complete this statement with a number. Area of circle = … # r (b) Lee drops 10 grains of rice at random onto the piece of paper. Diagram A shows the result. NOT TO SCALE A Lee removes the 10 grains of rice and drops another 10 grains of rice at random onto the piece of paper. Diagram B shows the result. NOT TO SCALE B

5 0607/53/M/J/18 © UCLES 2018 [Turn over (i) Complete the table. A B Combined results for all 20 grains of rice Number of grains of rice inside circle Relative frequency 10 20 (ii) Use the formula area of circle area of rectangle = relative frequency to estimate r using the combined results for all 20 grains of rice. r = …

6 0607/53/M/J/18 © UCLES 2018 3 Lee draws two circles on a different rectangular piece of paper. The circles touch the edges of the piece of paper and touch each other. x cm 8 cm 16 cm (a) (i) Find the value of x. … (ii) Complete this statement with a number. Total area of the two circles = … # r (b) Lee drops 50 grains of rice at random onto the piece of paper. He removes the 50 grains of rice and drops another 50 grains of rice at random onto the piece of paper. The table shows his results. First 50 Second 50 Combined results for all 100 grains of rice Number of grains of rice inside circle 36 43 Relative frequency Complete the table. (c) Use the formula total area of circles area of rectangle = relative frequency to estimate r. r = … (d) Give one reason why the estimate for r in question 3(c) is more accurate than the estimate for r in question 2(b)(ii). …

7 0607/53/M/J/18 © UCLES 2018 4 Lee draws one circle of radius r cm on a different piece of paper. The circle touches all four edges of the paper. r cm (a) Lee drops 500 grains of rice at random onto the piece of paper. He removes the 500 grains of rice and drops another 500 grains of rice at random onto the piece of paper. The combined number of grains of rice inside the circle is 785. Use the formula area of circle area of rectangle = relative frequency to estimate r. r = … (b) Complete this sentence with a single number. To estimate the value of r when the circle touches all four edges of the paper, multiply the relative frequency by … (c) Lee drops n grains of rice at random onto the piece of paper. He removes the n grains of rice and drops another n grains of rice at random onto the piece of paper. The combined number of grains of rice inside the circle is k. Show that n k 2 is an estimate for r.

8 0607/53/M/J/18 © UCLES 2018 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 International Examinations Copyright Acknowledgements Booklet. This is produced for each series of examinations and is freely available to download at www.cie.org.uk after the live examination series. Cambridge International Examinations is part of the Cambridge Assessment Group. Cambridge Assessment is the brand name of University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. BLANK PAGE

IGCSE™ is a registered trademark. This document consists of 5 printed pages. © UCLES 2018 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/53 Paper 5 (Core) May/June 2018 MARK SCHEME Maximum Mark: 24 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 2018 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 2018 © UCLES 2018 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 2018 © UCLES 2018 Page 3 of 5 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 2018 © UCLES 2018 Page 4 of 5 Question Answer Marks Partial Marks 1(a)(i) 800 1 C opportunity 1(a)(ii) [ ] 78.5 3 800 … 1 1(b)(i) 2 1 1(b)(ii) 2 10 oe isw 1 1(c) 3 20 1 1(d) 25 π 0.15 40 20 × = × oe M1 40 20 π 0.15 25 × = × oe M1 2(a) 100 1 2(b)(i) 4 6 10 cao 1 2(b)(ii) 3 2 M1 for 100 π 10 30 20 20 their × = × oe C opportunity 3(a)(i) 32 1 C opportunity 3(a)(ii) 128 1 C opportunity 3(b) 79 79 100 oe 1 3(c) 3.16 2 M1 for 128 π 79 16 100 their their their x × = × C opportunity 3(d) Valid explanation e.g. more grains of rice used 1 4(a) 3.14 nfww 3 M2 for 2 π 785 2 2 1000 r r r × = × or 2 2 0.785 4 r r π = × oe or M1 for 2 π 2 2 r r r × × oe or 785 1000 or 0.785 oe seen C opportunity

0607/53 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 5 of 5 Question Answer Marks Partial Marks 4(b) 4 1 4(c) 2 2 2 4 r k n r π = or 4 2 k n× oe 1 C opportunity Communication: seen in four of the following questions 2 1 for communication seen in three questions 1(a)(i) 40 × 20 seen 1(a)(ii) 0.0981[…] 2(b)(ii) for a further step of working e.g. cancelling or their 10 100π 30 20 20 = × × or 600 × 0.5 = 300 or = 100π or 600 0.5 1 00 their × 3(a)(i) for 4 × 8 oe 3(a)(ii) for 2 × 8 × 8 oe 3(c) for a further step e.g. 79 128π 32 16 100 their = × × 4(a) for a further step e.g. 785 4 1000 × or cancelling r or 0.785 4 π = oe or πr2 = 0.785 × 2r × 2r oe 4(c) for relative frequency is 2 k n seen