Cambridge IGCSE Mathematics - International 0607 — 2018 May/June Paper 6 · Variant 3

* 5 4 9 9 6 4 2 8 4 5 * This document consists of 14 printed pages and 2 blank pages. DC (SC/CGW) 153613/2 © UCLES 2018 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2018 1 hour 30 minutes 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 both parts A (Questions 1 to 4) and B (Questions 5 to 9). You must show all the 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 40.

2 0607/63/M/J/18 © UCLES 2018 Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 4) ESTIMATING r (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation is about using relative frequency to estimate the value of r. Area, A, of circle radius r is A = rr2. 1 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. (a) Lee draws a circle of radius 5 cm on a rectangular piece of paper measuring 40 cm by 20 cm. 5 cm 20 cm 40 cm NOT TO SCALE The probability, p, that a grain of rice lands inside the circle is p = area of circle area of paper . (i) Show that p is approximately 0.098 for this piece of paper. (ii) 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 0607/63/M/J/18 © UCLES 2018 [Turn over 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. … (iii) 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 . (iv) 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 r 25 # . Use area of circle area of paper = 0.15 to show that an estimate for r is 4.8 .

4 0607/63/M/J/18 © UCLES 2018 (b) 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 (i) Complete this statement with a number. Area of circle = … # r (ii) 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 0607/63/M/J/18 © UCLES 2018 [Turn over Complete the table. A B Combined results for all 20 grains of rice Number of grains of rice inside circle Relative frequency 20 10 (iii) Use the formula area of circle area of paper = relative frequency to estimate r using the combined results for all 20 grains of rice. r = …

6 0607/63/M/J/18 © UCLES 2018 2 (a) Lee draws a circle of radius 12 cm on a different piece of paper. The circle touches all four edges of the paper. 12 cm 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 combined number of grains of rice inside the circle is 78. Use the formula area of circle area of paper = relative frequency to estimate r. r = … (b) Lee draws a circle of radius r cm on a different piece of paper. The circle touches all four edges of the paper. r cm Show that, for any value of r, estimate for r = k # relative frequency, where k is an integer. Find the value of k. k = …

7 0607/63/M/J/18 © UCLES 2018 [Turn over 3 Lee draws a circle on a piece of paper in the shape of a regular hexagon of side length x cm. The circle touches all six edges of the paper. x cm NOT TO SCALE — x cm 2 √3 Lee drops grains of rice at random onto the piece of paper. He counts the number of grains of rice inside the circle and finds the relative frequency. (a) Find an estimate for the value of r when x = 30 and the relative frequency that the grain of rice is inside the circle is 0.905 . Use the formula area of circle area of paper = relative frequency . …

8 0607/63/M/J/18 © UCLES 2018 (b) Show that, for any value of x, estimate for r = k # relative frequency, where k is an exact constant. Find the value of k. k = …

9 0607/63/M/J/18 © UCLES 2018 [Turn over 4 Lee draws a circle on a piece of paper in the shape of a regular polygon with n sides. The circle touches all n edges of the paper. He drops grains of rice at random onto the piece of paper. He counts the number of grains of rice inside the circle and finds the relative frequency. Estimate for r = k # relative frequency Explain clearly why the constant k gives an approximation for r, as the value of n increases.

10 0607/63/M/J/18 © UCLES 2018 B MODELLING (QUESTIONS 5 to 9) SHOE BUSINESS (20 marks) You are advised to spend no more than 45 minutes on this part. This task is about the profit made by making and selling shoes. A company makes and sells two styles of shoe, X and Y. The objective of the task is for the company to maximise the amount of profit it makes each day. The number of shoes made each day is modelled using inequalities. Each day the company • makes x pairs of shoes of style X and y pairs of shoes of style Y • makes at most 150 pairs of shoes. 5 (a) (i) Write down, in terms of x and y, an inequality to show how many pairs of shoes can be made each day. … Two machines are used to make each pair of shoes. Machine A is used for at least 3 hours each day. Machine B is used for at least 4 hours each day. The table shows some information about making each style of shoe. Style X Style Y Maximum number of pairs of shoes made each day 80 100 Time taken by Machine A for each pair of shoes 5 min 2 min Time taken by Machine B for each pair of shoes 4 min 5 min Cost of making one pair of shoes $20 $15 Selling price of one pair of shoes $100 $70 (ii) Explain why x 0 80 G G and y 0 100 G G . … … (iii) An inequality for the total time each day, in minutes, that Machine A is used is x y 5 2 180 H + . Find, in terms of x and y, an inequality for the total time each day, in minutes, that Machine B is used. … (b) The company sells all the pairs of shoes it makes each day. Find an expression, in terms of x and y, for the total profit made each day. …

11 0607/63/M/J/18 © UCLES 2018 [Turn over 6 (a) Three lines x 80 = , y 100 = and x y 5 2 180 + = have been drawn on the grid. Draw two more lines to find the region defined by the five inequalities from question 5(a). Shade the unwanted region. 0 0 20 40 60 80 100 20 40 60 80 100 120 10 30 50 70 90 110 140 130 150 10 30 50 70 90 120 140 110 130 150 y x (b) Show that the greatest profit each day is $10 250. Write down the number of pairs of shoes of each style to make this profit. Style X … Style Y …

12 0607/63/M/J/18 © UCLES 2018 7 The table shows the amount of material for each style. Style X Style Y Material for one pair of shoes 0.5 m2 0.4 m2 There is a shortage of material and 50 m2 is used each day to make all the shoes. (a) Draw a line on the grid in question 6 to show this information. (b) Work out the decrease in the greatest profit because of the shortage. …

13 0607/63/M/J/18 © UCLES 2018 [Turn over 8 The company is considering making bags. Research shows that a model for the profit each day, $p, from the sale of b bags is . p b b 30 5 65 2 = - - . (a) On the axes below, sketch the graph of . p b b 30 5 65 2 = - - for b 0 35 G G . p b 0 35 (b) Find the minimum number of bags the company needs to sell each day in order to make a positive profit. … (c) The greatest number of bags the company can make each day is 80. Find the profit that the company will make each day when it sells all 80 bags. …

14 0607/63/M/J/18 © UCLES 2018 9 The company decides to make 80 bags each day. For every 2 bags that it makes, it has to decrease the number of pairs of style X shoes by one. The shortage of material for making shoes is the same as in question 7. There is no shortage of material for making bags. (a) Find the greatest number of pairs of shoes of style Y that the company can make. … (b) Show that the company’s profit each day has increased.

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16 0607/63/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 8 printed pages. © UCLES 2018 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2018 MARK SCHEME Maximum Mark: 40 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.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 2 of 8 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.

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 3 of 8 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

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 4 of 8 Question Answer Marks Partial Marks A INVESTIGATION ESTIMATING π 1(a)(i) 2 π 5 40 20 × × or π 25 800 × or 78.5 800 1 C opportunity 1(a)(ii) 2 10 oe isw 1 1(a)(iii) 3 20 1 1(a)(iv) 40 20 π 0.15 25 × = × oe 1 C opportunity 1(b)(i) 100 cao 1 1(b)(ii) 4 6 10 cao 1 1(b)(iii) 3 2 M1 for 100 π 10 oe 30 20 20 their × = × C opportunity 2(a) 3.12 2 M1 for 2 12 π 78 24 24 100 × = × oe C opportunity 2(b) [ ] 2 π relative frequency 2 2 r r r × = × M1 [k =] 4 B1 3(a) 3.134 to 3.138 or 3.14 or 3.13 nfww 2 M1 for area of hexagon = 2 3 3 30 2 × oe soi 2340 or awrt 2338 C opportunity 3(b) [ ] 2 2 3 π 4 relative frequency oe 3 3 2 x x × = M1 k = 2 3 B1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 5 of 8 Question Answer Marks Partial Marks 4 Relative frequency approaches 1 oe 1 Justification e.g. Almost all the discs will land inside the circle. or The [area of the] polygon approaches the [area of the] circle. 1 Communication: Seen in three of the following questions 2 1 for communication seen in two questions 1(a)(i) for 0.0981[...] 1(a)(iv) for 25 π 0.15 40 20 × = × or 25 × 4.8 = 120 or 0.15 × 800 = 120 or 120 0.15 800 = 1(b)(iii) for 10 30 20 π 20 100 their × = × or cancelling out factors such as 10 or 20 or 10 20 × 600 = 300 or 10 20 × 600 = their100 π 2(a) for 78 24 24 π 100 12 12 × = × × oe or π = 0.78oe × 4 or π 0.78 4 = oe

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 6 of 8 Question Answer Marks Partial Marks 3(a) for 2 2 3 30 π 2 3 3 30 2   ×     × oe or 2 2 3 3 0.905 30 2 π 3 30 2 × × =   ×     oe or 2 3 30 2   ×       π = 2 3 3 0.905 30 2 × × or 2120 or awrt 2116 to 2118 oe

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 7 of 8 Question Answer Marks Partial Marks B MODELLLING SHOE BUSINESS 5(a)(i) [0 ⩽ ] x + y ⩽ 150 1 5(a)(ii) The maximum of style X is 80 and the maximum of style Y is 100 and neither can be negative oe 1 5(a)(iii) 4x + 5y ⩾ 240 1 C opportunity 5(b) 80x + 55y oe 1 C opportunity 6(a) Correct ruled lines and region 4 FT their 5(a)(i) B1 for a line from (0, their 150) to (their 150, 0) FT their 5(a)(iii) B2 for a ruled line from (0, their 48) to (their 60, 0) or B1 for a ruled line through either (0, their 48) or (their 60, 0), or for an unruled line from (0, their 48) to (their 60, 0) or for a ruled line from (0, their 60) to (their 48, 0) FT their two lines B1 for their correct region 6(b) 80 × 80 + 55 × 70 = 10 250 soi 80, 70 1 7(a) Line from (0, 125) to (100, 0) 2 B1 for 0.5 0.4 50 x y + = oe 7(b) [$]2475 1 FT a line of 0.4x + 0.5y = 50 in (a) giving an answer of 1870 C opportunity 8(a) Correct sketch 1 Parabola, correct way up Negative y-intercept x-intercept between 30 and 35 C opportunity 8(b) 33 1 C opportunity 8(c) [$]3895 1 9(a) 75 1 C opportunity

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2018 © UCLES 2018 Page 8 of 8 Question Answer Marks Partial Marks 9(b) Finding 80(40) + 55(their 75) + their 3895 M1 soi by 11 220 their 11220 and compares to their 7775 A1 C Communication seen in any 3 of 2 C1 for communication seen in any 2 5(a)(iii) for 4 × 60 = 240 5(b) for 100 – 20 = 80 or 70 – 15 = 55 seen 7(b) for 7775 seen or x = 80, y = 25 soi or 45 × 55 in FT case, for 8380 seen or x = 80, y = 36 soi or 34 × 55 8(a) for marking a relevant scale or the y-intercept at −65 or the y co-ordinate of the minimum point marked at approximately 300 8(b) for 32.5; may be seen on graph or b = 32, p = –17 and b = 33, p = 17.5 9(a) for attempt to solve 0.5 × 40 + 0.4y = 50 oe or FT case: attempt to solve 0.4 × 40 + 0.5y = 50 oe or for line drawn on graph at x = 40 in 6(a)