Cambridge IGCSE Mathematics - International 0607 — 2016 May/June Paper 6 · Variant 1

This document consists of 12 printed pages. DC (LK/FD) 115851/4 © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education * 5 6 5 4 9 5 8 8 7 8 * CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) May/June 2016 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 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 0607/61/M/J/16 © UCLES 2016 Answer both parts A and B. A INVESTIGATION MOVING TRIANGLES (20 marks) You are advised to spend no more than 45 minutes on this part. Q P A R S 1 B This investigation is about finding the connection between AP and BQ as P and Q move. All triangles are right-angled and RS is one unit. In questions 1, 2 and 3, P is at A. 1 R Q P S B Triangle PBQ is an enlargement of triangle PSR, with P as the centre of the enlargement. (a) Write down the scale factor of the enlargement in the diagram. … (b) Complete the table for enlarging triangle PSR. Scale factor Length of PS Length of PB 3 4 6 30 7 14 (c) Use one word to complete this statement. PSR and PBQ are … triangles because one is an enlargement of the other.

3 0607/61/M/J/16 © UCLES 2016 [Turn over 2 P S B Q NOT TO SCALE x R 2 1 (a) Show that x = 10 when PB = 20. (b) Find the value of x when PB = 16. … (c) Find an expression for x when PB = y. … 3 P S R x 1 Q B NOT TO SCALE 4 y For this triangle, find an expression for x when PB = y. …

4 0607/61/M/J/16 © UCLES 2016 4 These diagrams show P starting at A and then moving towards B. (a) P S B Q R 3 1 6 NOT TO SCALE y In this diagram P is at A. Find the value of y. … (b) P A S B Q R 2 1 6 NOT TO SCALE z In this diagram, P and S have moved towards B. PS is one unit less than in part (a). Find the value of z. … (c) Using your answers to part (a) and part (b), work out the value of AP. …

5 0607/61/M/J/16 © UCLES 2016 [Turn over 5 P S R Q B x y 5 1 A P S R Q B x z 4 1 NOT TO SCALE These diagrams show P starting at A and moving towards B. In the second diagram, PS has decreased by one unit. Using the method of question 4, show that AP = BQ.

6 0607/61/M/J/16 © UCLES 2016 6 P S R Q B x y n 1 A P S R Q NOT TO SCALE B x z 1 These diagrams show P starting at A and moving towards B. In the second diagram, PS has decreased by one unit. Show that AP = BQ.

7 0607/61/M/J/16 © UCLES 2016 [Turn over 7 In this question, RS is no longer one unit. P starts at A and moves towards B. In the second diagram, PS has decreased by one unit. P S R Q B x y n P A S R Q NOT TO SCALE B x z (a) When RS = 2, find an expression for AP in terms of x. … (b) When RS = m, find an expression for AP in terms of x and m. …

8 0607/61/M/J/16 © UCLES 2016 B MODELLING MUSICAL NOTES (20 Marks) You are advised to spend no more than 45 minutes on this part. This task is about the connection between musical notes. A musical note is made by a sound wave which is modelled by a sine function. Here is the sine wave for the note A1, where the time, t, is measured in seconds. 1 t 0 –1 1 55 2 55 Each note has a different frequency, which is measured in Hertz (Hz). The frequency of the note A1 is 55 Hz because the sine wave repeats 55 times per second. 1 The frequency of the note A2 is two times the frequency of the note A1. On the grid below, sketch the sine wave for the note A2 for t 0 55 2 G G . 1 t 0 –1 1 55 2 55

9 0607/61/M/J/16 © UCLES 2016 [Turn over 2 The 12 notes in a musical scale are A, A#, B, C, C#, D, D#, E, F, F#, G, G#. The notes on a piano repeat this scale. Notes in the same scale have the same subscript. (For example, A2, C#2 and F2 are all in the same scale.) The frequency, f Hz, of each note on a piano is modelled by the function ( ) . f n 27 5 2 n 12 # = where n is an integer from 0 to 87. n 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 87 Note A0 A#0 B0 C0 C#0 D0 D#0 E0 F0 F#0 G0 G#0 A1 A#1 B1 (a) When n = 0 the frequency of the note is 27.5 . This is the note A0. (i) Work out the frequency of the note when n = 3. … (ii) Write down the note when n = 15. … (iii) Work out the frequency of the note E0. … (b) Write down all the values of n that give the note A on this piano. … (c) Which note has the highest frequency on this piano? Calculate this frequency. Note … Frequency …

10 0607/61/M/J/16 © UCLES 2016 3 k times the frequency of a note gives the frequency of the next note. This means that ( ) ( ) f f k n n 1 = + . Find the exact value of k. … 4 (a) On the axes below, sketch the graph of ( ) . g x 27 5 2 x 12 # = for x 0 87 G G . 5000 87 g (x) x 0 (b) Find the note which has frequency closest to 1400 Hz. …

11 0607/61/M/J/16 © UCLES 2016 [Turn over 5 A different musical scale has 10 notes. Q R S T U V W X Y Z The frequency of each note is modelled by the function ( ) h n a 2bn # = where n is an integer from 0 to 29. When n = 0 the note is Q0 and the frequency of this note is 600 Hz. The frequency of the note Q1 is 1200 Hz. (a) Write down the value of a. … (b) Find the value of b. … (c) Show that ( ) ( ) h h k n n 1 = + where k is a constant to be found. Question 6 is printed on the next page.

12 0607/61/M/J/16 © UCLES 2016 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. 6 In musical scales the frequency of the note P1 is two times the frequency of the note P0. (a) A musical scale has 23 notes. The frequency of the first note is 75 Hz. Work out the frequency of the second note. … (b) The first note in another musical scale has a frequency of 100 Hz. The second note has a frequency of 108 Hz. Find the number of notes in this scale. …

® IGCSE is the registered trademark of Cambridge International Examinations. This document consists of 5 printed pages. © UCLES 2016 [Turn over Cambridge International Examinations Cambridge International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/61 Paper 6 (Extended) May/June 2016 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 will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2016 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 2016 0607 61 © Cambridge International Examinations 2016 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 MOVING TRIANGLES Question Answer Mark Part Marks 1 (a) 2 1 (b) Scale factor PS PB 3 4 12 5 6 30 7 2 14 3 B1 for each one correct (c) Similar 1 2 (a) 2 1 20 10 = oe 1 Allow, for example, 2 : 20 = 1 : 10 or 2 : 1 = 20 : 10 or 2 × 10 = 20 and 1 × 10 = 10 or 2: 20 and 1 : x so 2x = 20, x = 10 or PS is double RS so PB is double QB or equivalent (b) 8 1 C opportunity (c) 2 y oe 1 condone 2 y × 1 3 4 y oe 1 condone 4 y × 1 If 0 scored in 2(c) and 3, allow SC1 for answers of y = 2x and y = 4x

Page 3 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 61 © Cambridge International Examinations 2016 Question Answer Mark Part Marks 4 (a) 18 1 C opportunity (b) 12 1 C opportunity (c) their 6 1FT strict FT their y − their z 5 [y =] 5x and [z =] 4x [AP =] 5x – 4x = x M1 A1 may be on diagram Allow 2 marks for y = 5x and z = 4x seen or clearly indicated [AP =] y – z = x 6 [AP =] nx – (n – 1)x = x 1 or nx – (nx – x) = x or nx – nx + x = x not from wrong working or equating expressions for BQ 1 y z n n = − and rearranging to show that either y y z n − = with y x n = or that 1 z y z n − = − with 1 z x n = − C opportunity 7 (a) 2 x 2 M1 for 1 2 xn and 1 ( 1) 2 x n − oe seen or for x = 2AP (b) x m 1 C opportunity Communication seen in 3 of 2(b), 4(a), 4(b), 6 or 7(b) 2 C1 if seen in two of them

Page 4 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 61 © Cambridge International Examinations 2016 B MODELLING MUSICAL NOTES Question Answer Mark Part Marks 1 Correct curve over full domain. 2 B1 for at least one correct, complete cycle e.g. over the domain 1 0 110 t

or for a graph of incorrect shape but that has 4 cycles over the full domain or for a graph with more than 3 inaccurate t-intercepts with 4 cycles over the full domain or for a fully correct and accurate sketch graph of the sine wave for the note A0 2 (a) (i) 32.7[0] or 32.703 to 32.7032 isw 1 (ii) C1 1 (iii) 41.2[0] or 41.203 to 41.2035 isw 1 C opportunity (b) [0, 12,] 24, 36, 48, 60, 72, 84 1 (c) C7 and 4190 or 4186 or 4186.0 or 4186.00 or 4186.009 to 4186.01 1 3 1 12 2 or exact equivalent 1 isw conversion to decimal, but decimal answer only does not score C opportunity 4 (a) Correct exponential shape 1 Intent of smooth curve; must not cross x-axis; condone graph not drawn on full domain; condone y-intercept at origin; (b) F5 2 M1 for n = 68 soi e.g. f(68) or 68 12 27.5 2 × C opportunity -1 1 t 0 1 55 2 55

Page 5 Mark Scheme Syllabus Paper Cambridge IGCSE – May/June 2016 0607 61 © Cambridge International Examinations 2016 Question Answer Mark Part Marks 5 (a) 600 1 (b) 1 10 oe isw 1 (c) Uses an algebraic process to find either 1 10 h( 1) 2 h( ) their n n + = × oe or k = 1 10 2 their or 1.07 or 1.071 to 1.072 1FT FT their value of b, provided b ≠1; Allow 2b k = isw Condone k found by calculating the ratio of at least 2 pairs of consecutive values e.g. h(2) h(1) and h(4) h(3) 6 (a) 77.3 or 77.29 to 77.295 2 M1 for 23 2 k where k may be a constant or a variable seen C opportunity (b) 9 2 not from wrong working M1 for 100 2 108 n × = or 100 1.08 200 n × = or 1.08 2 n = or for 1.089 = 1.99… soi or for two correct trials using a valid relationship seen C opportunity Communication in 2 of 2(a)(iii), 3, 4(b), 6(a) or 6(b) 2 C1 if seen in 1 of them