Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 Oct/Nov Paper 2 · Variant 3

This document consists of 12 printed pages. IB11 11_0581_23/RP © UCLES 2011 [Turn over *0585912066* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/23 Paper 2 (Extended) October/November 2011 1 hour 30 minutes Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Mathematical tables (optional) Tracing paper (optional) 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. You may use a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, glue or correction fluid. DO NOT WRITE IN ANY BARCODES. Answer all questions. If working is needed for any question it must be shown below that question. Electronic calculators should be used. If the degree of accuracy is not specified in the question, and if the answer is not exact, give the answer to three significant figures. Give answers in degrees to one decimal place. For π, use either your calculator value or 3.142. At the end of the examination, fasten all your work securely together. The number of marks is given in brackets [ ] at the end of each question or part question. The total of the marks for this paper is 70. www.XtremePapers.com

2 © UCLES 2011 0581/23/O/N/11 For Examiner's Use 1 Martha divides $240 between spending and saving in the ratio spending : saving = 7 : 8 . Calculate the amount Martha has for spending. Answer $ [2] 2 210 211 212 213 214 215 216 From the list of numbers, find (a) a prime number, Answer(a) [1] (b) a cube number. Answer(b) [1] 3 Solve the simultaneous equations. x + 5y = 22 x + 3y = 12 Answer x = y = [2]

3 © UCLES 2011 0581/23/O/N/11 [Turn over For Examiner's Use 4 Find the value of 3 4 8 27 −       . Give your answer as an exact fraction. Answer [2] 5 The population of a city is 128 000, correct to the nearest thousand. (a) Write 128 000 in standard form. Answer(a) [1] (b) Write down the upper bound of the population. Answer(b) [1] 6 Pedro invested $800 at a rate of 5% per year compound interest. Calculate the total amount he has after 2 years. Answer $ [2] 7 Show that 3 –2 + 2 –2 = 36 13 . Write down all the steps of your working. Answer [2]

4 © UCLES 2011 0581/23/O/N/11 For Examiner's Use 8 Find the value of 8.36 10.4 1.89 17.1 3 + − . Answer [2] 9 In Vienna, the mid-day temperatures, in °C, are recorded during a week in December. This information is shown below. –2 2 1 –3 –1 –2 0 Calculate (a) the difference between the highest temperature and the lowest temperature, Answer(a) °C [1] (b) the mean temperature. Answer(b) °C [2] 10 Maria decides to increase her homework time of 8 hours per week by 15%. Calculate her new homework time. Give your answer in hours and minutes. Answer h min [3]

5 © UCLES 2011 0581/23/O/N/11 [Turn over For Examiner's Use 11 Factorise completely. p2x – 4q2x Answer [3] 12 Alberto changes 800 Argentine pesos (ARS) into dollars ($) when the rate is $1 = 3.8235 ARS. He spends $150 and changes the remaining dollars back into pesos when the rate is $1 = 3.8025 ARS. Calculate the amount Alberto now has in pesos. Answer ARS [3] 13 During a marathon race an athlete loses 2 % of his mass. At the end of the race his mass is 67.13 kg. Calculate his mass before the race. Answer kg [3]

6 © UCLES 2011 0581/23/O/N/11 For Examiner's Use 14 r 2r NOT TO SCALE The sphere of radius r fits exactly inside the cylinder of radius r and height 2r. Calculate the percentage of the cylinder occupied by the sphere. [The volume, V, of a sphere with radius r is V = 3 4 πr3.] Answer % [3] 15 ap = px + c Write p in terms of a, c and x. Answer p = [3]

7 © UCLES 2011 0581/23/O/N/11 [Turn over For Examiner's Use 16 The time, t, for a pendulum to swing varies directly as the square root of its length, l. When l = 9, t = 6 . (a) Find a formula for t in terms of l. Answer(a) t = [2] (b) Find t when l = 2.25 . Answer(b) t = [1] 17 R F In the Venn diagram, = {students in a survey}, R = {students who like rugby} and F = {students who like football}. n( ) = 20 n(R ∪ F) = 17 n(R) = 13 n(F) = 11 (a) Find (i) n(R ∩ F), Answer(a)(i) [1] (ii) n(RV∩ F=). Answer(a)(ii) [1] (b) A student who likes rugby is chosen at random. Find the probability that this student also likes football. Answer(b) [1]

8 © UCLES 2011 0581/23/O/N/11 For Examiner's Use 18 Write as a single fraction, in its simplest form. x x x x 2 1 2 1 − + − − Answer [4] 19 O A B 9 cm 50° NOT TO SCALE The diagram shows a sector AOB of a circle, centre O, radius 9 cm with angle AOB = 50°. Calculate the area of the segment shaded in the diagram. Answer cm2 [4]

9 © UCLES 2011 0581/23/O/N/11 [Turn over For Examiner's Use 20 (a) N =       6 2 . The order of the matrix N is 2 × 1. P = (1 3). The order of the matrix P is 1 × 2. (i) Write down the order of the matrix NP. Answer(a)(i) [1] (ii) Calculate PN. Answer(a)(ii) [1] (b) M =       4 2 3 2 . Find M–1, the inverse of M . Answer(b) M–1 = [2]

10 © UCLES 2011 0581/23/O/N/11 For Examiner's Use 21 North North A C B 4.5 km 2.7 km 5 km NOT TO SCALE The diagram shows 3 ships A, B and C at sea. AB = 5 km, BC = 4.5 km and AC = 2.7 km. (a) Calculate angle ACB. Show all your working. Answer(a) Angle ACB = [4] (b) The bearing of A from C is 220°. Calculate the bearing of B from C. Answer(b) [1]

11 © UCLES 2011 0581/23/O/N/11 [Turn over For Examiner's Use 22 A B C D X 4.40 cm 3.84 cm 9.40 cm NOT TO SCALE A, B, C and D lie on a circle. AC and BD intersect at X. (a) Give a reason why angle BAX is equal to angle CDX. Answer(a) [1] (b) AB = 4.40 cm, CD = 9.40 cm and BX = 3.84 cm. (i) Calculate the length of CX. Answer(b)(i) CX = cm [2] (ii) The area of triangle ABX is 5.41 cm2. Calculate the area of triangle CDX. Answer(b)(ii) cm2 [2] Question 23 is printed on the next page.

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 2011 0581/23/O/N/11 For Examiner's Use 23 0 40 Speed (m / s) Time (minutes) 5 10 15 NOT TO SCALE The diagram shows the speed-time graph for the first 15 minutes of a train journey. The train accelerates for 5 minutes and then continues at a constant speed of 40 metres/second. (a) Calculate the acceleration of the train during the first 5 minutes. Give your answer in m/s2. Answer(a) m/s2 [2] (b) Calculate the average speed for the first 15 minutes of the train journey. Give your answer in m/s. Answer(b) m/s [3]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2011 question paper for the guidance of teachers 0581 MATHEMATICS 0581/23 Paper 2 (Extended), maximum raw mark 70 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 must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the October/November 2011 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 23 © University of Cambridge International Examinations 2011 Abbreviations cao correct answer only cso correct solution only dep dependent ft follow through after error isw ignore subsequent working oe or equivalent SC Special Case www without wrong working Qu. Answers Mark Part Marks 1 112 2 M1 for 240 ÷ (7 + 8) × 7 2 (a) 211 cao 1 (b) 216 cao 1 3 (x =) –3 (y =) 5 2 M1 for correctly eliminating one variable 4 81 16 cao 2 B1 for16 81 , 81 k , k 16 or (2/3)4 seen 5 (a) 1.28 × 105 1 (b) 128 500 1 6 882 2 M1 800 × 1.05 × 1.05 7 9 1 , 4 1 M1 Both fractions seen       = + 4 1 9 1 36 4 + 36 9 = 36 13 E1 Both fractions over a common denominator and added to give 36 13 8 0.186 2 B1 for 2.477 to 2.478 or 13.29… seen 9 (a) 5 or −5 1 (b) −0.714 (−0.7143 to −0.7142) or −7 5 2 M1 for −2 + 2 + 1 − 3 − 1 − 2 and ÷ 7 10 9 h 12 min 3 M1 for 8 × 1.15 A1 for 9.2 B1 ft independent for their 9.2 correctly converted into hours and minutes 11 x(p – 2q)(p + 2q) 3 M2 for (px – 2qx)(p + 2q) or (p – 2q)(px + 2qx) or M1 for x(p2 – 4q2) 12 225.(23112) 3 M2 for (800 ÷ 3.8235 – 150) × 3.8025 M1 for 800 ÷ 3.8235 13 68.5 www 3 M2 for 67.13 ÷ 0.98 or M1 for 67. 13 is 98% 14 3 2 66 or 66.7 www 3 M2 for ) 2 ( π π 2 3 3 4 r r r (× 100) or M1 for ) 2 ( π 2 r r 15 x a c p − = 3 M1 one correct move M1 second correct move M1 third correct move marked on answer line

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 23 © University of Cambridge International Examinations 2011 16 (a) l t 2 = 2 M1 for l k t = (b) 3 1ft Ft dependent on using l k t = 17 (ii) 7 1 (ii) 4 1 (b) 13 7 oe 1ft Ft their Venn diagram or their (a)(i)/13 18 ( ) x x x x 2 1 5 1 2 − + − or 2 2 2 5 1 x x x x − + − 4 M1 for (1 – x)(1 – 2x) – x(2 + x) seen B1 for 1 – x – 2x + 2x2 or 1 – 3x + 2x2 seen B1 for x(1 – 2x) oe as a common denominator 19 4.32 4 M1 for 2 9 360 50 × π × M1 for 0.5 × 92 × sin 50 M1 for subtracting their triangle from their sector (dependent on at least M1) 20 (a) (i) 2 × 2 1 (ii) (20) 1 Brackets essential (b)       − − 2 2 3 4 2 1 oe 2 M1 for       d c b a 2 1 or       − − 2 2 3 4 k seen 21 (a) 84(.00..) 4 M2 for cos (…) = 5.4 7.2 2 5 5.4 7.2 2 2 2 × × − + or (M1 for 52 = 2.72 + 4.52 – 2 × 2.7 × 4.5 × cos C ) A1 for 0.1045… (implied by correct answer) (b) 136 1ft 220 – their (a) 22 (a) Angles in same segment 1 (b) (i) 8.2(0) 2 M1 for 4.4 4.9 84 .3 = CX (= 2.136) oe (ii) 24.7 2 M1 for 2 4.4 4.9 41 .5       = ∆ (= 4.564) oe 23 (a) 0.133 (3…) or 15 2 2 M1 for 40 ÷ 300 seen (b) 33 3 1 or 33.3 3 M1 for area under graph attempted M1 for correct total area statement