Cambridge IGCSE Mathematics - International 0607 — 2019 May/June Paper 6 · Variant 3

* 7 0 9 6 7 7 4 5 4 9 * This document consists of 16 printed pages. DC (NH/CGW) 170705/2 © UCLES 2019 [Turn over CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2019 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 3) and B (Questions 4 to 6). 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. Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education

2 0607/63/M/J/19 © UCLES 2019 The Investigation starts on page 3.

3 0607/63/M/J/19 © UCLES 2019 [Turn over Answer both parts A and B. A INVESTIGATION (QUESTIONS 1 to 3) JUMPING FROGS (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the number of different ways that a frog can jump between stones in a line. The stones are always 1 unit apart. The frog always jumps • from left to right • from the first stone to the last stone. The diagram shows a frog sitting on a stone in a pond. First stone Last stone This frog has a jump length of 1 unit. This is enough to move from one stone to the next stone. There is only 1 way to jump between two stones. There is only 1 way to jump between three stones. First stone Last stone When there are more than three stones in a line, there is always only 1 way for this frog to jump from the first stone to the last stone. 1 A different frog has a maximum jump length of 2 units. There is still only 1 way to jump between two stones. There are now 2 ways to jump between three stones.

4 0607/63/M/J/19 © UCLES 2019 (a) Complete the diagrams below to show the 3 ways that this frog can jump between four stones. (b) Complete the diagrams below to show the 5 ways that this frog can jump between five stones.

5 0607/63/M/J/19 © UCLES 2019 [Turn over (c) The table shows the number of ways to jump between stones when the maximum jump length is 2 units. Number of stones Number of ways 2 1 3 2 4 3 5 5 6 8 7 13 8 9 The numbers in the last column of the table form a sequence. (i) Complete the table. (ii) Write down the rule to find further terms in this sequence. … …

6 0607/63/M/J/19 © UCLES 2019 2 Another frog has a maximum jump length of 3 units. There is only 1 way for this frog to jump between two stones. There are 2 ways to jump between three stones. (a) There are 4 ways to jump between four stones. These are the same 3 ways as in question 1(a) and 1 new way. Draw the new way on the diagram below. (b) There are the same 5 ways as in question 1(b) to jump between five stones and some new ways. Draw diagrams to show the new ways below.

7 0607/63/M/J/19 © UCLES 2019 [Turn over (c) The table shows the number of ways to jump between stones when the maximum jump length is 3 units. Number of stones Number of ways 2 1 3 2 4 4 5 6 13 7 24 8 9 The numbers in the last column of the table form a sequence. (i) Use your answer to part (b) to help you complete the table. (ii) Write down the rule to find further terms in this sequence. … …

8 0607/63/M/J/19 © UCLES 2019 3 The table shows the number of ways to jump between 2 to 9 stones when the maximum jump length is 1 to 8 units. Number of stones Maximum jump length 1 unit 2 units 3 units 4 units 5 units 6 units 7 units 8 units 2 1 1 1 1 1 1 1 1 3 1 2 2 2 2 2 2 2 4 1 3 4 4 4 4 4 4 5 1 5 8 8 8 8 8 6 1 8 13 15 16 16 16 16 7 1 13 24 29 31 32 32 32 8 1 56 61 63 64 64 9 1 120 125 127 128 (a) Complete the table. (b) The numbers in the column for maximum jump length of 4 units form a sequence. Write down the rule for finding further terms in this sequence. … …

9 0607/63/M/J/19 © UCLES 2019 [Turn over (c) The terms of the sequence for a maximum jump length of 8 units are 1, 2, 4, 8, 16, 32, 64, 128 (i) Find an expression, in terms of x, for the number of ways of jumping between x stones, where . x 2 9 G G … (ii) Show that this expression does not give the number of ways for . x 10 = (d) Complete the two rules for finding the number of ways of jumping between x stones when the maximum jump length is m units. Rule 1 For x 2 G G … the number of ways = … Rule 2 For x > … the number of ways is found by … (e) A frog sits on the first of 20 equally spaced stones. The frog has a maximum jump length of 18 units. Find the number of ways the frog can jump between the 20 stones. …

10 0607/63/M/J/19 © UCLES 2019 B MODELLING (QUESTIONS 4 to 6) PLAY TENTS (20 marks) You are advised to spend no more than 45 minutes on this part. This task is about maximising the volume of a tent for a given area of material. A company makes play tents for children. They make them in the shape of cuboids or square-based pyramids. The top and sides of the tents are made from material. No material is used for the base. Cuboid tent Pyramid tent 4 The cuboid tent has a square base of side x metres and height h metres. h m x m x m (a) Write down the volume, V, of the cuboid in terms of h and x. … (b) Show that the total area, Am2, of material for the cuboid tent is . A x hx 4 2 = +

11 0607/63/M/J/19 © UCLES 2019 [Turn over (c) The company uses 8m2 of material to make each cuboid tent. Use this information and part (a) and part (b) to show that a model for the volume is ( ) V x x 4 8 2 = - . (d) (i) On the axes below, sketch the graph of the model ( ) V x x 4 8 2 = - . V 0 3 x (ii) Find the maximum volume of a cuboid tent made from 8m2 of material. … (e) Customers want to buy tents with a height of at least 1 metre. (i) Using your answer to part (a), sketch, on the axes above, the graph of V against x when h = 1. (ii) Find the volume of a cuboid tent made from 8m2 of material with a height of 1 metre. …

12 0607/63/M/J/19 © UCLES 2019 5 The pyramid tent also has a square base of side x metres and a height of h metres. The distance between the top of the pyramid and the midpoint of each edge of the base is y metres. h m y m x m The triangle below shows one of the faces of the pyramid. x m y m (a) Find a model, in terms of x and y, for the total area, S, of the four triangular faces of the pyramid. Give your answer in its simplest form. …

13 0607/63/M/J/19 © UCLES 2019 [Turn over (b) Find an expression for y2 in terms of h and x. (c) Use part (a) and part (b) to show that h x S x 4 4 2 2 2 = - c m.

14 0607/63/M/J/19 © UCLES 2019 (d) The company uses 8m2 of material to make each tent. Volume of a pyramid 3 1 # = base area # height (i) Show that a model for the volume of the pyramid tent is V x x x 3 16 4 2 2 2 = - c m. (ii) On the axes below, sketch the graph of V x x x 3 16 4 2 2 2 = - c m. V 0 3 x

15 0607/63/M/J/19 © UCLES 2019 [Turn over (iii) Write down the maximum volume of a pyramid tent that uses 8m2 of material. … (iv) Customers want to buy tents with a height of at least 1 metre. Sketch, on the axes on page 14, the graph of V against x when . h 1 = (v) Find the maximum volume of a pyramid tent made from 8m2 of material with a height of at least 1 metre. … Question 6 is printed on the next page.

16 0607/63/M/J/19 © UCLES 2019 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. 6 The company decides that the cuboid tent and the pyramid tent should have the same volume. Which type of tent will customers want to buy? Give a reason for your answer. Type of tent … because … …

This document consists of 9 printed pages. © UCLES 2019 [Turn over Cambridge Assessment International Education Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/63 Paper 6 (Extended) May/June 2019 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 2019 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 2019 © UCLES 2019 Page 2 of 9 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 2019 © UCLES 2019 Page 3 of 9 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 2019 © UCLES 2019 Page 4 of 9 Question Answer Marks Partial Marks A INVESTIGATION JUMPING FROGS 1(a) 1 1(b) 2 B1 for any two correct 1(c)(i) 21 34 1 C opportunity 1(c)(ii) add the previous two terms oe 1 2(a) 1 2(b) 1 2(c)(i) 7 (13) (24) 44 81 1 C opportunity 2(c)(ii) add the previous three terms oe 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 5 of 9 Question Answer Marks Partial Marks 3(a) 108 in ‘4 units’ column 1 C opportunity 3(b) add the previous four terms oe 1 3(c)(i) 2x – 2 oe 2 B1 for recognition of powers of 2 3(c)(ii) 256 and sum of previous 8 terms = 255 oe 1 3(d) m + 1 2x–2 m + 1 summing the previous m terms. 3 B1 for m + 1 (only) in either place B1 for 2x–2 B1 for ‘summing previous m terms’ oe 3(e) 262143 or 218 – 1 final answer 2 B1 for 2n – 1, n integer > 8 Communication: seen in one of the following questions 1 1(c)(i) 8 + 13 or 13 + their21 2(c)(i) their7 + 13 + 24 or 13 + 24 + their44 3(a) 8 + 15 + 29 + 56 oe 3(c)(ii) 10 substituted in 2their algebraic expression

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 6 of 9 Question Answer Marks Partial Marks B MODELLING PLAY TENTS 4(a) hx2 oe 1 4(b) top = x2 oe [4] sides = [4]hx oe 1 4(c) 2 8 4 x h x − = M1 C opportunity 2 2 8 4 x V x x   − =     or V = 2 2 (8 ) 4 x x x − M1 OR 4hx2 = (8 – x2) x M1 Correct rearrangements leading to x2 + 4xh = 8 M1 OR V = 2 2 ( 4 ) 4 x xh x x + − M1 Correct simplification leading to V = x2h M1 4(d)(i) Correct sketch 1 Curve through (0, 0) with x-intercept < 3. C opportunity 4(d)(ii) 2.18 or 2.177… 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 7 of 9 Question Answer Marks Partial Marks 4(e)(i) Correct sketch 1 4(e)(ii) 2.14[...] 1 C opportunity 5(a) S = 2xy 1 C opportunity 5(b) 2 2 2 x h   +     oe 1 5(c) 2 2 2 4 x h y = − M1 Substitute 2 S y x = oe M1 OR Substitute S2 = 4x2y2 M1 Simplify leading to 2 2 2 4 x h y = − M1 5(d)(i) Substitution of h into V = 1 3 x2h and substitution of S = 8 leading to given answer 1

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 8 of 9 Question Answer Marks Partial Marks 5(d)(ii) Correct sketch 1 Curve through (0, 0) with x-intercept < 3 implied. Maximum to right of x = 1.5 C opportunity 5(d)(iii) 2.34 or 2.339… 1 C opportunity 5(d)(iv) Correct sketch 1 5(d)(v) 2.34 or 2.339… 1 FT their 5(d)(iii) C opportunity 6 Pyramid, [because] the cuboid tent of that volume will have a height <1 m oe. 1 C opportunity

0607/63 Cambridge IGCSE – Mark Scheme PUBLISHED May/June 2019 © UCLES 2019 Page 9 of 9 Question Answer Marks Partial Marks Communication: seen in five of the following questions 2 C1 for three opportunities seen 4(c) x2 + 4xh = 8 oe or 4xh = 8 – x2 4(d)(i) Scale with maximum slightly above 2 or maximum labelled 4(d)(ii), 4(e)(ii) m3 in either 5(a) 1 2 × x × y oe 5(d)(ii) Scale with maximum slightly above 2 or maximum labelled 5(d)(iii). 5(d)(v) m3 in either 6 Sketch or identify volume as 2.16.