Cambridge IGCSE Mathematics - International 0607 — 2014 May/June Paper 6 · Variant 1

This document consists of 8 printed pages. IB14 06_0607_61/5RP © UCLES 2014 [Turn over *5450797540* Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) May/June 2014 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 © UCLES 2014 0607/61/M/J/14 Answer both parts A and B. A INVESTIGATION COUNTING FACTORS (20 marks) You are advised to spend 45 minutes on part A. This investigation looks for a method to find how many factors a number has. 1 (a) (i) Write down, in ascending order, the five factors of 16. Two have been written down for you. 1 , , , , 16 (ii) Write down the five factors of 16 as powers of 2. Two have been written down for you. 20 , , , , 24 (b) Write down, in ascending order, the four factors of 27 as powers of 3. , , , 2 (a) p is a prime number. (i) Write down the six factors of p5 as powers of p. One has been written down for you. p0 , , , , , (ii) Write down, in terms of n, the number of factors of pn. (b) 78 125 = 5n where 5 is a prime number. Find the number of factors of 78 125. Do not write them out.

3 © UCLES 2014 0607/61/M/J/14 [Turn over 3 (a) 20 is not a prime number. 20 = 22 × 51 where 2 and 5 are prime numbers. Find all the factors of 20 by completing the table. Powers of 5 50 51 Powers of 2 20 20 × 50 = 1 × 1 = 1 20 × 51 = …… × …… = …… 21 21 × 50 = 2 × 1 = 2 21 × 51 = …… × …… = …… 22 22 × 50 = …… × …… = …… 22 × 51 = 4 × 5 = 20 (b) The table has 3 rows and 2 columns. Describe how to find the number of factors of 20 from the number of rows and the number of columns. (c) 4000 = 25 × 53. Find the number of factors of 4000. 4 (a) (i) 1 000 000 = 2n × 5n. Find n. (ii) Find the number of factors of 1 000 000. (b) Find n such that 10n has 900 factors.

4 © UCLES 2014 0607/61/M/J/14 5 x has exactly two prime factors, 3 and 7. So x can be written as 3m × 7n. Find all the possible values of x that have exactly 12 factors. 6 (a) Write 336 in the form pd × qe × rf where p, q and r are prime numbers. × × (b) Find the number of factors of 336. 7 A number has exactly three prime factors, 2, 3 and 5. It has exactly 12 factors. Find all possible numbers.

5 © UCLES 2014 0607/61/M/J/14 [Turn over B MODELLING TIDES (20 marks) You are advised to spend 45 minutes on part B. In this task you will model the height of a tide. Here is the graph of y = sin x°. Its period is 360. y x 30 60 90 120 150 180 210 240 270 300 330 360 1 0.5 –0.5 –1 0 1 (a) (i) On the grid below, sketch the graph of y = sin(3x)°. y x 30 60 90 120 150 180 210 240 270 300 330 360 1 0.5 –0.5 –1 0 (ii) Write down the period of the graph of y = sin(3x)°. (b) Write down the period of the graph of y = sin(10x)°. (c) Write down an expression, in terms of b, for the period of the graph of y = sin(bx)°.

6 © UCLES 2014 0607/61/M/J/14 2 The graph below shows the approximate height of the tide at Auckland (New Zealand) during 25th April 2014. t is the time of day starting at midnight. h is the height in metres above a fixed height. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1.5 1 0.5 –0.5 –1 –1.5 0 h t (a) Write down the period of the graph. (b) Explain why a model for the height of the tide on 25th April is h = 1.2 sin(30t)°. (c) Usually the height of a tide is measured as the height, H metres, above sea level. The graph below shows this for the tide at Auckland on 25th April. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 4 3 2 1 0 H t Modify the model in part (b). H =

7 © UCLES 2014 0607/61/M/J/14 [Turn over (d) The best fishing conditions are when the height of the tide is less than one metre above sea level. (i) Use your calculator with your model for H to find the two times between which there were the best fishing conditions on the morning of 25th April. Give your times correct to the nearest minute. and (ii) Using your answer to part (i), write down the times between which there were the best fishing conditions on the afternoon of 25th April. and 3 Here is a different model for the height of the tide, D metres, on the morning of 25th April. D = 0.022t3 – 0.403t2 + 1.9t (a) Compare D with H from question 2(c) by drawing their graphs on your calculator for 0 < t < 12. Comment on how the difference in height between D and H varies with time. (b) Make one change to the model D that makes the graph of D very close to the graph of H for 3 < t < 9. D = Question 4 and 5 are printed on the next page.

8 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. 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. © UCLES 2014 0607/61/M/J/14 4 The graph below shows the actual height of the tide above sea level on 25th April 2014 at Auckland. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 4 3 2 1 0 H t The actual period for the height of a tide is 12 hours 25 minutes. Use this fact to improve your model for the height of the tide above sea level on 25th April by changing one of the numbers in your answer to question 2(c). H = 5 The graph for 26th April can be found by translating the graph for 25th April horizontally. Modify your answer to question 4 to give a model for the height of the tide above sea level at Auckland on 26th April. H =

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2014 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 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 2014 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components.

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 61 © Cambridge International Examinations 2014 A INVESTIGATION COUNTING FACTORS 1 (a) (i) (ii) (b) 2, 4, 8 2[1], 22, 23 30, 3[1], 32, 33 1 1 1 2 (a) (i) (ii) (b) p[1], p2, p3, p4, p5 n + 1 8 1 1 1 C opportunity 3 (a) (b) (c) Powers of 5 50 51 Powers of 2 20 20 × 50 = 1 × 1 = 1 20 × 51 = 1 × 5 = 5 21 21 × 50 = 2 × 1 = 2 21 × 51= 2 × 5 = 10 22 22 × 50 = 4 × 1 = 4 22 × 51= 4 × 5 = 20 multiply oe 24 1 1 1 must not be part of incorrect statement C opportunity 4 (a) (i) (ii) (b) 6 soi 49 29 1 1FT 1 FT their (6 + 1)2, their 6 ≠ 0, 1 C opportunity C opportunity 5 1323 1701 3087 50421 3 B2 for 3 numbers seen or for 4 numbers seen with extras or B1 for 2 or 1 numbers seen if 0 scored then SC2 for 3[1] × 75, 32 × 73, 33 × 72 and 35 × 7[1] and no extras or SC1 for 3[1] × 75, 32 × 73, 33 × 72 and 35 × 7[1] with extras C opportunity

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 61 © Cambridge International Examinations 2014 6 (a) (b) 24 × 3[1] × 7[1] isw 20 1 1FT FT only if indices are three positive integers soi C opportunity 7 60 90 150 2 B1 for 2 or 1 correct numbers If 0 scored M1 for 2² × 3 × 5, 2 × 3² × 5, 2 × 3 × 5² C opportunity Communication seen in 3 of the following: 2(b), 3(c), 4(a)(ii), 4(b), 5, 6(b), 7 1

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 61 © Cambridge International Examinations 2014 B MODELLING TIDES 1 (a) (i) (ii) (b) (c) 120 36 360 b or 360 : b 2 1 1 1 B1 correct maximum and minimum B1 correct period over their domain 2 (a) (b) (c) (d) (i) (ii) 12 [hours] amplitude or maximum = 1.2 soi 360 12 soi or 360 12 30 = soi 1.2sin(30t)° + 2 isw [0]7 53 or 07 54 10 07 or 10 06 19 53 22 07 1 2 1 2 1FT B1 for each B1 for each or M1 for 7.8 to 8[.0] and 10[.0] to 10.2 seen if 0 scored then SC1 for 473 and 607 minutes as final answer C opportunity FT their times + 12 3 (a) (b) Any two valid comments about the difference in height over a range of time 0.022t3 ‒ 0.403t2 + 1.9t + 0.4 1 + 1 1 For example: [more] close between 4 and 11 oe not close before 4 and after 11 oe SC1 for a comment such as H is higher than D until about 11 when they are the same and then D is higher than H

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0607 61 © Cambridge International Examinations 2014 4 1.2sin(29t)° + 2 2FT FT from 2(c) M1 for 5 12 360 12 oe or 360 12.4[...] oe or 360 12.4[...] b = oe SC1 1.2sin(29.4t)° + 2 C opportunity 5 1.2sin(their29(t – 5 6 ))° + 2 or 1.2sin(their29t – 24.2)° + 2 isw 2FT FT from 4 B1 for 5 6 or 0.83 oe seen If 0 scored then SC1 if inside brackets missing or SC1 for t – 5 12 or t – 0.416 or t – 0.8[…] in the full expression. Communication seen in one of the following questions: 2 (d), 4 1