Cambridge IGCSE Mathematics (with coursework) 0581 — 2010 May/June Paper 2 · Variant 1

This document consists of 12 printed pages. IB10 06_0581_21/FP © UCLES 2010 [Turn over *3935169314* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/21 Paper 2 (Extended) May/June 2010 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 2010 0581/21/M/J/10 For Examiner's Use 1 Write the numbers in order of size with the smallest first. 10 3.14 22 7 π Answer < < < [2] 2 Michel changed $600 into pounds (£) when the exchange rate was £1 = $2.40. He later changed all the pounds back into dollars when the exchange rate was £1 = $2.60. How many dollars did he receive? Answer $ [2] 3 p is the largest prime number between 50 and 100. q is the smallest prime number between 50 and 100. Calculate the value of p – q. Answer [2] 4 A person in a car, travelling at 108 kilometres per hour, takes 1 second to go past a building on the side of the road. Calculate the length of the building in metres. Answer m [2]

3 © UCLES 2010 0581/21/M/J/10 [Turn over For Examiner's Use 5 Calculate the value of 5(6 × 103 + 400), giving your answer in standard form. Answer [2] 6 Calculate the value of 1 1 1 1 2 2 2 2 + (a) writing down all the figures in your calculator answer, Answer(a) [1] (b) writing your answer correct to 4 significant figures. Answer(b) [1] 7 0.8 m 1.4 m NOT TO SCALE The top of a desk is made from a rectangle and a quarter circle. The rectangle measures 0.8m by 1.4m. Calculate the surface area of the top of the desk. Answer m2 [3]

4 © UCLES 2010 0581/21/M/J/10 For Examiner's Use 8 (a) Shade one square in each diagram so that there is (i) one line of symmetry, [1] (ii) rotational symmetry of order 2. [1] (b) The pyramid below has a rectangular base. The vertex of the pyramid is vertically above the centre of the base. Write down the number of planes of symmetry for the pyramid. Answer(b) [1]

5 © UCLES 2010 0581/21/M/J/10 [Turn over For Examiner's Use 9 A cyclist left Melbourne on Wednesday 21 May at 09 45 to travel to Sydney. The journey took 97 hours. Write down the day, date and time that the cyclist arrived in Sydney. Answer Day Date Time [3] 10 3.5 m 1.5 m NOT TO SCALE The diagram represents a rectangular gate measuring 1.5m by 3.5m. It is made from eight lengths of wood. Calculate the total length of wood needed to make the gate. Answer m [3]

6 © UCLES 2010 0581/21/M/J/10 For Examiner's Use 11 Make d the subject of the formula 5 4 2 d w c w + = . Answer d = [3] 12 Q = {2, 4, 6, 8, 10} and R = {5, 10, 15, 20}. 15 ∈ P, n(P) = 1 and P ∩ Q = Ø. Label each set and complete the Venn diagram to show this information. [3] 13 Solve the simultaneous equations. 2 7 2 x y + = 2 17 2 x y − = Answer x = y = [3]

7 © UCLES 2010 0581/21/M/J/10 [Turn over For Examiner's Use 14 y varies inversely as the square of x. y = 1.5 when x = 8. Find y when x = 5. Answer y = [3] 15 The points (2, 5), (3, 3) and (k, 1) all lie in a straight line. (a) Find the value of k. Answer(a) k = [1] (b) Find the equation of the line. Answer(b) [3]

8 © UCLES 2010 0581/21/M/J/10 For Examiner's Use 16 Simplify (a) 0.75 4 16 p       , Answer(a) [2] (b) 32q-3 ÷ 23q-2. Answer(b) [2] 17 AB is the diameter of a circle, centre O. C, D and E lie on the circle. EC is parallel to AB and perpendicular to OD. Angle DOC is 38°. Work out (a) angle BOC , Answer(a) Angle BOC = [1] (b) angle CBO, Answer(b) Angle CBO = [1] (c) angle EDO . Answer(c) Angle EDO = [2] O A E D C B 38° NOT TO SCALE

9 © UCLES 2010 0581/21/M/J/10 [Turn over For Examiner's Use 18 y x 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 A B C D E F H G Write down the letters of all the triangles which are (a) congruent to the shaded triangle, Answer(a) [2] (b) similar, but not congruent, to the shaded triangle. Answer(b) [2]

10 © UCLES 2010 0581/21/M/J/10 For Examiner's Use 19 The position vector r is given by r = 2p + t(p + q). (a) Complete the table below for the given values of t. Write each vector in its simplest form. One result has been done for you. t 0 1 2 3 r 4p + 2q [3] (b) O is the origin and p and q are shown on the diagram. (i) Plot the 4 points given by the position vectors in the table. O q p [2] (ii) What can you say about these four points? Answer(b)(ii) [1]

11 © UCLES 2010 0581/21/M/J/10 [Turn over For Examiner's Use 20 f(x) = (x – 1)3 g(x) = (x – 1)2 h(x) = 3x + 1 (a) Work out fg(-1). Answer(a) [2] (b) Find gh(x) in its simplest form. Answer(b) [2] (c) Find f -1(x). Answer(c) [2] Question 21 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 2010 0581/21/M/J/10 For Examiner's Use 21 (a) A is a (2 × 4) matrix, B is a (3 × 2) matrix and C is a (1 × 3) matrix. Which two of the following matrix products is it possible to work out? A2 B2 C2 AB AC BA BC CA CB Answer(a) and [2] (b) Find the inverse of 1 3 2 4 1 1 8 4             . Simplify your answer as far as possible. Answer(b)           [3] (c) Explain why the matrix 4 2 6 3         does not have an inverse. Answer(c) [1]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2010 question paper for the guidance of teachers 0581 MATHEMATICS 0581/21 Paper 21 (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. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2010 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 – May/June 2010 0581 21 © UCLES 2010 Qu. Answers Mark Part Marks 1 3.14 π 7 22 √10 2 M1 3.1428(…) and 3.16(2…) seen 2 650 2 M1 4.2 600 (× 2.6) 3 44 2 M1 97 or 53 seen 4 30 2 M1 108 × 1000 / (60 × 60) 5 3.2(0) × 104 2 B1 32000 or 32 × 103 etc 6 (a) 0.461939(…) (b) 0.4619 or ft 1 1ft 7 1.62 3 M1 4 1 π 0.82 M1 adding (0.8 × 1.4) to their k π 8 (a) (i) 1 or (ii) 1 (b) 2 1 9 Sunday (May) 25 1045 1, 1, 1 Independent 10 24.3(0788…) 3 M1 5 × 3.5 + 2 × 1.5 M1 (√) 1.52 + 3.52 11 5 4 2 w cw − oe 3 M1 one correct move to clear fractions M1 second correct move to subtract term M1 third correct move dividing by 5 May be in any order 12 4 6 10 P 15 2 8 5 20 R Q 3 M1 15 only in small circle M1 10 only in the intersection A1 all correct including labels 13 x = 12 y = –10 3 M1 consistent addition (& mult) for x or consistent subtraction (& mult) for y A1 only earned if method correct 14 3.84 or 3 25 21 3 M1 y = 2 x k oe A1 k = 96

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0581 21 © UCLES 2010 15 (a) 4 1 (b) y = –2x + 9 oe 3 M1 3 2 3 5 − − oe M1 substitution of a point into their equation If M1 only then A1ft for y = “m”x + “c” used correctly with their numeric values 16 (a) 8 3 p or 0.125p3 1, 1 Independent marks for letter and no. (b) 8 9 q–1 1, 1 Independent marks for letter and no. Allow 1 8 1 q–1 or q 8 9 17 (a) 52 (b) 64 (c) 71 1 1 2 M1 angle CED = 19 18 (a) E, G (b) A, B 1, 1 1, 1 19 (a) 2p 3p + q ……….. 5p + 3q cao (b) (i) all 4 plotted correctly ft (ii) a (straight) line 1, 1, 1 2 1 B1 2 or 3 correct Allow linear, collinear 20 (a) 27 (b) 9x2 cao (c) 3√x + 1 2 2 2 M1 g(–1) = 4 seen or ((x – 1)2 – 1)3 M1 (3x + 1 – 1)2 or better M1 interchange x, y & rearrange formula 21 (a) CB and BA cao 1, 1 Independent (b)     − − 16 4 24 8 cao 3 M1 8 1 4 3 4 1 2 1 × − × (= 32 1 )= M1           − − 2 1 8 1 4 3 4 1 seen (c) determinant is zero 1 Allow cannot divide by zero