Cambridge IGCSE Mathematics - International 0607 — 2015 May/June Paper 6 · Variant 2

This document consists of 12 printed pages. DC (AC/CGW) 100447/1 © UCLES 2015 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 4 0 1 6 8 6 6 9 8 7 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended) May/June 2015 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 and B. 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 40.

2 0607/62/M/J/15 © UCLES 2015 Answer both parts A and B. A INVESTIGATION MOLECULES (20 marks) You are advised to spend no more than 45 minutes on this part. This investigation looks at the number of spheres and rods that you need to make models of molecules. 1 Chemists use small spheres and rods to make models of molecules. These diagrams show a sequence of molecules of height 1. Molecule 1 Molecule 2 Molecule 3 (a) Draw the next two molecules in this sequence. (b) Complete this table for molecules of height 1. Molecule m Number of spheres s Number of rods r 1 1 0 2 2 1 3 3 2 4 5 6 (c) Write down a formula for s in terms of m. s = …

3 0607/62/M/J/15 © UCLES 2015 [Turn over 2 These diagrams show a sequence of molecules of height 2. Molecule 1 Molecule 2 Molecule 3 Molecule 4 (a) Complete this table for molecules of height 2. Molecule m Number of spheres s Number of rods r 1 2 1 2 4 4 3 6 7 4 5 6 (b) Find, in terms of m, a formula for (i) s, s = … (ii) r. r = …

4 0607/62/M/J/15 © UCLES 2015 3 These diagrams show a sequence of molecules of height 3. Molecule 1 Molecule 2 Molecule 3 Molecule 4 (a) Complete this table for molecules of height 3. Molecule m Number of spheres s Number of rods r 1 3 2 2 6 7 3 9 12 4 5 6 (b) Find, in terms of m, a formula for (i) s, s = … (ii) r. r = …

5 0607/62/M/J/15 © UCLES 2015 [Turn over 4 (a) Use your answers to questions 1(c), 2(b) and 3(b) to help you complete the table for molecules of height h. Height (h) Number of spheres (s) in terms of m Number of rods (r) in terms of m 1 m − 1 2 3 4 5 5m 9m − 5 6 (b) Find, in terms of m and h, a formula for (i) s, s = … (ii) r. r = …

6 0607/62/M/J/15 © UCLES 2015 (c) Use your answer to part (b)(i) to find a formula for m in terms of s and h. m = … (d) Find a formula for r in terms of s and h. r = …

7 0607/62/M/J/15 © UCLES 2015 [Turn over 5 (a) A molecule has height h and width w. For example, Molecule 4 in question 3 has h = 3 and w = 4. Use your answer to question 4(d) to show that r = 2hw – h − w. (b) Can a square molecule have 544 rods?

8 0607/62/M/J/15 © UCLES 2015 B MODELLING WHERE IS THE HORIZON? (20 marks) You are advised to spend no more than 45 minutes on this part. This table shows the distance to the horizon (y kilometres) at different heights above sea level (x metres). x 1 2 3 4 5 6 7 8 9 10 y 3.6 5.0 6.2 7.1 8.0 8.7 9.4 10.1 10.7 11.3 0 1 1 2 3 4 5 6 Height above sea level (m) Distance to horizon (km) 7 8 9 10 x y 2 3 4 5 6 7 8 9 10 11 12

9 0607/62/M/J/15 © UCLES 2015 [Turn over 1 (a) On the grid, plot the points given in the table. The points (2, 5) and (5, 8) have been plotted for you. (b) The simplest way to model the data is with a straight line. (i) On the grid, draw the straight line passing through (2, 5) and (5, 8). Find the equation of this line. … (ii) Using this model what is the distance to the horizon when at sea level? …

10 0607/62/M/J/15 © UCLES 2015 2 Another model for the data is y ax bx c 2 = + + . Assume that, when the height is 0, the distance to the horizon is 0. (a) Show that c = 0. (b) When c = 0, the model is y ax bx 2 = + . (i) Use the point (2, 5) to form an equation in a and b. … (ii) Use the point (5, 8) to form another equation in a and b. …

11 0607/62/M/J/15 © UCLES 2015 (c) Solve the equations in part (b) and write down your model. … (d) Sketch the graph of y against x for x 0 10 G G . 0 x y (e) Comment on the validity of this model. … Question 3 is printed on the next page.

12 0607/62/M/J/15 © UCLES 2015 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. 3 Another model for this data is y axb = . (a) Use (2, 5) and (5, 8) to write down two equations in a and b. … … (b) Show that . . 1 6 2 5b = . (c) Show that b = 0.5, correct to 1 decimal place. (d) Find the value of a and write down your model. … (e) Compare the model in this question with the data on page 8. … …

® IGCSE is the registered trademark of Cambridge International Examinations. CAMBRIDGE INTERNATIONAL EXAMINATIONS Cambridge International General Certificate of Secondary Education MARK SCHEME for the May/June 2015 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/62 Paper 6 (Extended), maximum raw mark 40 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2015 series for most Cambridge IGCSE®, Cambridge International A and AS Level components and some Cambridge O Level components.

Page 2 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 62 © Cambridge International Examinations 2015 Abbreviations cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent SC Special Case nfww not from wrong working soi seen or implied A INVESTIGATION 1 (a) 1 (b) 4 3 5 4 6 5 1 (c) [s =] m 1 2 (a) 8 10 10 13 12 16 1 (b) (i) [s =] 2m oe 1 (ii) [r =] 3m – 2 oe 1 C opportunity 3 (a) 12 17 15 22 18 27 1 (b) (i) [s =] 3m oe 1 (ii) [r =] 5m – 3 oe 1 C opportunity

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 62 © Cambridge International Examinations 2015 4 (a) m m – 1 2m 3m – 2 3m 5m – 3 4m 7m – 4 5m 9m – 5 6m 11m – 6 2 B1 for row 4 B1 for row 6 If 0 scored, SC1 for one correct column of 6 items (b) (i) [s =] hm oe 1 (ii) [r =] (2h – 1)m – h oe isw 1 (c) [m = ] h s 1 (d) [r =] (2h – 1) h s – h oe isw 1FT FT substituting their 4(c) in their 4(b)(ii) 5 (a) h s = w oe r = (2h – 1)w – h 2 B1 can be implied by seeing substitution of w = h s or s = wh in their 4(d) B1 (b) Yes, if h = 17 (only) oe 2 M1 for 544 = 2h2 – 2h with attempt to solve by factorisation, formula, sketch, completing the square, approximation or trial and improvement with three improving trials If 0 scored, SC1 for 17 (without wrong working) or for Yes if 17 and –16 Communication seen in one of 2(b)(ii), 3(b)(ii), 5(b) 1

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 62 © Cambridge International Examinations 2015 B MODELLING 1 (a) 8 points correctly plotted 2 B1 for 6 or 7 correct (b) (i) y = x + 3 oe 2 M1 for m = 1 soi by, e.g. y = x (ii) 3 1 C opportunity 2 (a) 0 = 0[2] + 0 + c 1 (b) (i) 5 = 4a + 2b oe isw 1 (ii) 8 = 25a + 5b oe isw 1 (c) Equating coefficients soi or writing one equation correctly as a = or b = Combining their equations correctly to eliminate one variable or substitution of a or b a = –0.3 or b = 3.1 oe their second variable correct M1FT M1FT A1 B1FT FT their 2(b) if coefficients not equal dep on both method marks dep on one method mark FT their first variable in one of their equations in 2(b) If 0 scored, SC1 for a = –0.3 and b = 3.1 or correct model without working (d) Parabola through (0, 0) with local maximum seen 1 C opportunity (e) Not valid oe and y decreases soi by, e.g max = 8 or Valid oe for [0 <] x < 5 or less than max or Invalid oe for x > 5 or Not valid oe and negative oe 1 dep on mark in (d)

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2015 0607 62 © Cambridge International Examinations 2015 3 (a) 5 = a2b 8 = a5b isw 1 (b) b b a a 2 5 5 8 = soi 1 (c) log2.5 log1.6 or log2.51.6 or 2.50.513 = 1.6 or 2.5m = a value less than 1.6 with 2.5n = a value more than 1.6 1 2.5b = 1.6 and b = 0.513 0.45 ≤ m < 0.5125… with 0.5135… < n ≤ 0.55. (d) y = 3.5x0.5 oe 1 Model must be written in full (e) close fit or suitable oe 1 dep on model in (d) Communication seen in one of 1(b)(ii), 2(d) 1