Cambridge IGCSE Mathematics - International 0607 — 2013 Oct/Nov Paper 6 · Variant 1

This document consists of 11 printed pages and 1 blank page. IB13 11_0607_06/2RP © UCLES 2013 [Turn over *8778931706* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/06 Paper 6 (Extended) October/November 2013 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, highlighters, glue or correction fluid. You may use a 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. www.XtremePapers.com

2 © UCLES 2013 0607/06/O/N/13 For Examiner's Use Answer both parts A and B. A INVESTIGATION SUMS OF SEQUENCES (20 marks) You are advised to spend no more than 45 minutes on this part. Here is the method to construct a sequence for this investigation. Method Example Write down any two numbers for the first two terms. 3 and 7 Add these two terms to make the third term. 3 + 7 = 10 Add the second and third terms to make the fourth term. 7 + 10 = 17 Add the third and fourth terms to make the fifth term. 10 + 17 = 27 Continue in this way to construct the sequence. This example makes the sequence: 3, 7, 10, 17, 27, 44, 1 Show that the sum of the first six terms in this sequence, divided by the fifth term, is 4.

3 © UCLES 2013 0607/06/O/N/13 [Turn over For Examiner's Use 2 (a) The first two terms of another sequence are 4.52 and 16.9 . (i) Use the method to write down the next four terms in this sequence. Do not round any of your numbers. 4.52, 16.9, , , , (ii) Work out the sum of the first six terms in this sequence and divide it by the fifth term. (b) (i) Choose two negative numbers to be the first two terms of a sequence. Use the method to work out the next four terms in this sequence. Write down the first six terms in your sequence. , , , , , (ii) Work out the sum of the first six terms in your sequence and divide it by the fifth term.

4 © UCLES 2013 0607/06/O/N/13 For Examiner's Use 3 The first two terms of a new sequence are p and q. The table shows the working for the first five terms. Working Term p p q q p + q p + q q + p + q p + 2q p + q + p + 2q 2p + 3q (a) Complete the table. (b) Find an expression for the sum of these first six terms. Simplify your answer. (c) Find an equation to connect the fifth term and the sum of the first six terms.

5 © UCLES 2013 0607/06/O/N/13 [Turn over For Examiner's Use 4 (a) Find the next four terms in the sequence in question 3. 7th term 8th term 9th term 10th term (b) Find an expression for the sum of the first ten terms in this sequence. Simplify your answer. (c) Find an equation to connect the seventh term and the sum of the first ten terms.

6 © UCLES 2013 0607/06/O/N/13 For Examiner's Use 5 (a) Find the next four terms in the sequence in question 4. 11th term 12th term 13th term 14th term (b) Find an expression for the sum of the first fourteen terms. Simplify your answer. (c) This sum is a multiple of one of the terms in question 4(a). Find this multiple. (d) Prove this connection.

7 © UCLES 2013 0607/06/O/N/13 [Turn over For Examiner's Use 6 In question 2 the connection between the sum of the six terms and the fifth term is sum of the first 6 terms = 4 times 5th term Complete the following statements. sum of the first 10 terms = sum of the first 14 terms = sum of the first 18 terms =

8 © UCLES 2013 0607/06/O/N/13 For Examiner's Use B MODELLING THE EARTH’S TEMPERATURE (20 marks) You are advised to spend no more than 45 minutes on this part. Logarithms to base 10 are written as log. Scientists have been measuring the temperature of the earth since 1860. The table shows the increase in Earth’s temperature since 1860. The increases are averages for each decade (10 years) correct to 2 decimal places. Final year of each decade Number of years since 1860 (N) Temperature increase since 1860 in °C (T) 1890 30 0.02 1900 40 0.03 1910 50 0.04 1920 60 0.06 1930 70 0.08 1940 80 0.10 1950 90 0.13 1960 100 0.18 1970 110 0.24 1980 120 0.32 1 (a) On the grid, plot the temperature increase (T) against the number of years since 1860 (N), for 30 Ğ N Ğ 120. Draw a smooth curve that shows the increase in temperature. 0.4 0.3 0.2 0.1 0 40 30 50 60 70 80 Number of years since 1860 90 100 110 120 130 T N Temperature increase (°C)

9 © UCLES 2013 0607/06/O/N/13 [Turn over For Examiner's Use (b) (i) Which of the following models best fits this graph? T = aN + b T = asinbN T = aN b T = | aN + b | (ii) Use the values of N and T for 1900 and 1940 in your model to write down two equations in a and b. (iii) Use your equations to show that 0.3 = 0.5b. (iv) Find the value of b and show that it rounds to 1.74, correct to 3 significant figures. (v) Find the value of a. Give your answer correct to 2 significant figures. a = (vi) Show that your model gives a suitable value of T for 1920.

10 © UCLES 2013 0607/06/O/N/13 For Examiner's Use 2 (a) (i) Complete the table to give the value of log T for each value of N. Give each answer correct to 2 decimal places. Final year of each decade Number of years since 1860 (N) Temperature increase since 1860 in °C (T) log T 1890 30 0.02 –1.70 1900 40 0.03 1910 50 0.04 1920 60 0.06 –1.22 1930 70 0.08 1940 80 0.10 1950 90 0.13 –0.89 1960 100 0.18 1970 110 0.24 –0.62 1980 120 0.32 –0.49 (ii) On the grid plot log T against N, for 30 Ğ N Ğ 120. 0.5 0 –0.5 –1 –1.5 –2 –2.5 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 N log T

11 © UCLES 2013 0607/06/O/N/13 For Examiner's Use (iii) The mean point is (75, –1.07). On the grid, draw the line of best fit. (iv) Use your line of best fit to predict the temperature increase by 2020 (160 years since 1860). (b) (i) A model for the line of best fit is logT = mN + c. Find the values of m and c. Give your answers correct to 2 significant figures. m = c = (ii) Use this model to predict the temperature increase by 2020. (iii) Comment on your two predictions for the temperature increase by 2020.

12 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. University of 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 2013 0607/06/O/N/13 BLANK PAGE

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2013 series 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/06 Paper 40 (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 October/November 2013 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components. www.XtremePapers.com

Page 2 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0607 06 © Cambridge International Examinations 2013 A INVESTIGATION SUMS OF SEQUENCES 1 108 ÷ 27 [= 4] 1 2 (a) (i) 21.42, 38.32, 59.74, 98.06 1 (ii) 4 www 1FT FT their total ÷ their 5th number (b) (i) Candidate’s own negative sequence correct 1 (ii) 4 www 1 Dep on (b)(i) correct 3 (a) p + 2q + 2p + 3q │ 3p + 5q 1,1 Accept different order (b) 8p + 12q oe isw or 5p + 7q plus their 3p + 5q or 4 times 5th term 1FT FT their 6th term in 3(a) C opportunity (c) 4 12 8 3 2 q p q p + = + OR 8p + 12q = 4(2p + 3q) isw OR 4 3 2 12 8 = + + q p q p 1 4 (a) 5p + 8q 8p + 13q 13p + 21q 21p + 34q 2FT FT their previous 6th term in p and q in 3(a) B1 for any two correct including after incorrect FT If 0 scored SC1 for explicit sum of 2 previous terms not totalled for all 4 correct (b) 55p + 88q oe isw 1 C opportunity (c) 11 88 55 8 5 q p q p + = + OR 11(5p + 8q) = 55p + 88q isw OR 11 8 5 88 55 = + + q p q p 1

Page 3 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0607 06 © Cambridge International Examinations 2013 5 (a) 34p + 55q 55p + 89q 89p + 144q 144p + 233q 2FT FT their previous 9th and 10th terms in p and q in 4(a) B1 for any two correct including after incorrect FT If 0 scored SC1 for explicit sum of 2 previous terms not totalled for all 4 correct (b) 377p + 609q oe isw 1 C opportunity (c) 29 soi 1 C opportunity (d) 377p + 609q = 29 ( 13p + 21q) seen oe 1 SC1 if this statement seen in (c) and not here 6 [sum of first 10 terms = ] 11 times 7th term [sum of first 14 terms = ] 29 times 9th term [sum of first 18 terms = ] 76 times 11th term 1 Communication seen in one of 3(b) 4(b) 5(b) 5(c) 1 Total 20

Page 4 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0607 06 © Cambridge International Examinations 2013 B MODELLING THE EARTH’S TEMPERATURE 1 (a) 10 correctly plotted points ± 1mm N P2 D1 P1 for 8 or 9 correct points ± 1 mm D1 for smooth curve through plotted points (b) (i) T = aNb 1 (ii) 0.03 = a [×] 40b 0.1 = a [×] 80b isw 1 (iii) (0.03 = a × 40b) ÷ (0.1 = a × 80b) oe isw 1 (iv) [b = ] 1.73696… correct to at least 3dp 1 C opportunity (v) [a = ] (4.88… to 4.95…) × 10–5 1 C opportunity (vi) T = (4.9 × 10–5) × N1.74 Substitute N = 60 to give T ≈ 0.06 (0.0606 – 0.0609) isw 1FT FT their a in part (v) 2 (a) (i) Number of years since 1860 (N) Temperature Increase °C (T) log T 30 0.02 −1.70 40 0.03 −1.52 50 0.04 −1.4[0] 60 0.06 −1.22 70 0.08 −1.1[0] 80 0.10 −1[.00] 90 0.13 −0.89 100 0.18 −0.74 110 0.24 −0.62 120 0.32 −0.49 2 –1 eeoo

Page 5 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0607 06 © Cambridge International Examinations 2013 (ii) 9 correctly plotted points ± 2 mm P1 (iii) 1 Line (within tolerance) through mean (within –1.05 to –1.1) (iv) FT from their line of best fit in part (iii) Correct to 1dp 2FT M1 reading log T correctly from their graph ± 2 mm Line must reach 160 If 0 scored in (iii) allow M1 only (b) (i) [m =] 0.006... to 0.018… [c =] –2.4… to –1.7… 1 1 If 0 scored M1 for working using 2 points on the line C opportunity (ii) FT from their m and c in (i), substituted in model Accuracy to 1dp 1FT C opportunity (iii) Comment on 2020 being outside range of given data 1 Communication seen in one of 1(b)(iv) 1(b)(v) 2(b)(i) 2(b)(ii) 1 Total 20