Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 May/June Paper 3 · Variant 1

This document consists of 16 printed pages. IB11 06_0581_31/2RP © UCLES 2011 [Turn over *6367778450* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/31 Paper 3 (Core) May/June 2011 2 hours 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 104. www.XtremePapers.com

2 © UCLES 2011 0581/31/M/J/11 For Examiner's Use 1 Mr and Mrs Clark and their three children live in the USA and take a holiday in Europe. (a) Mr Clark changes $500 into euros (€) when the exchange rate is €1 = $1.4593. Calculate how much he receives. Give your answer correct to 2 decimal places. Answer(a) € [2] (b) Tickets for an amusement park cost €62 for an adult and €52 for a child. Work out the cost for Mr and Mrs Clark and their three children to visit the park. Answer(b) € [3] (c) Mr Clark sees a notice: SPECIAL OFFER! Family ticket €200 Work out €200 as a percentage of your answer to part (b). Answer(c) % [1]

3 © UCLES 2011 0581/31/M/J/11 [Turn over For Examiner's Use (d) Mrs Clark buys 6 postcards at €0.98 each. She pays with a €10 note. Calculate how much change she will receive. Answer(d) € [2] (e) Children under a height of 130 cm are not allowed on one of the rides in the park. Helen Clark is 50 inches tall. Use 1 inch = 2.54 cm to show that she will not be allowed on this ride. Answer(e) [1]

4 © UCLES 2011 0581/31/M/J/11 For Examiner's Use 2 The shape above is the net of a solid drawn on a 1 cm square grid. (a) Write down the geometrical name of the solid. Answer(a) [1] (b) Find the perimeter of the net. Answer(b) cm [1]

5 © UCLES 2011 0581/31/M/J/11 [Turn over For Examiner's Use (c) Work out (i) the area of one of the triangles, Answer(c)(i) cm2 [2] (ii) the volume of the solid. Answer(c)(ii) cm3 [2] (d) A cuboid of length 4 cm and width 3 cm has the same volume as the solid. Calculate the height of the cuboid. Answer(d) cm [2]

6 © UCLES 2011 0581/31/M/J/11 For Examiner's Use 3 (a) x = 3m – k Find the value of (i) x when m = 2 and k = −4, Answer(a)(i) [2] (ii) m when x = 19 and k = 5. Answer(a)(ii) [3] (b) Expand the brackets. g(7f – g2) Answer(b) [2] (c) Factorise completely. 18h2 – 12hj Answer(c) [2] (d) Make m the subject of the formula. t = 8m + 15 Answer(d) m = [2] (e) Solve the equation. p + 3 = 3(p – 5) Answer(e) p = [3]

7 © UCLES 2011 0581/31/M/J/11 [Turn over For Examiner's Use 4 5 4 3 2 1 0 0 20 40 60 Time (minutes) Distance (kilometres) 80 100 120 140 Home Bus stop Library Sonia travels from home to the library. She walks to the bus stop and waits for a bus to take her to the library. (a) Write down (i) the distance to the bus stop, Answer(a)(i) km [1] (ii) how many minutes Sonia waits for a bus, Answer(a)(ii) min [1] (iii) how many minutes the bus journey takes to the library. Answer(a)(iii) min [1] (b) Calculate, in kilometres per hour, (i) Sonia’s walking speed, Answer(b)(i) km/h [1] (ii) the speed of the bus, Answer(b)(ii) km/h [2] (iii) the average speed for Sonia’s journey from home to the library. Answer(b)(iii) km/h [3] (c) Sonia works in the library for one hour. Then she travels home by car. The average speed of the car is 30 km/h. Complete the travel graph. [2]

8 © UCLES 2011 0581/31/M/J/11 For Examiner's Use 5 y x 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –3 –2 –1 0 1 2 3 4 A B (a) (i) Find the gradient of the line AB. Answer(a)(i) [2] (ii) Write down the equation of the line AB in the form y = mx + c. Answer(a)(ii) y = [2]

9 © UCLES 2011 0581/31/M/J/11 [Turn over For Examiner's Use (b) The table shows some values of the function y = x2 O 2. x −3 −2 −1 0 1 2 3 y 7 −1 −1 7 (i) Complete the table. [2] (ii) On the grid, draw the graph of y = x2 O 2 for −3 Y x Y 3. [4] (iii) Use your graph to solve the equation x2 O 2 = 0. Answer(b)(iii) x = or x = [2] (c) Write down the co-ordinates of the points where your graph meets the line AB. Answer(c)( , ) and ( , ) [2]

10 © UCLES 2011 0581/31/M/J/11 For Examiner's Use 6 (a) 103 112 125 132 144 159 161 From the list above, write down (i) a square number, Answer(a)(i) [1] (ii) a cube number, Answer(a)(ii) [1] (iii) a prime number, Answer(a)(iii) [1] (iv) an odd number which is a multiple of 3. Answer(a)(iv) [1] (b) Write 88 as a product of prime numbers. Answer(b) [2] (c) Find the highest common factor of 72 and 96. Answer(c) [2] (d) Find the lowest common multiple of 15 and 20. Answer(d) [2]

11 © UCLES 2011 0581/31/M/J/11 [Turn over For Examiner's Use 7 (a) B A P T (i) Reflect triangle T in the line AB. Label your image X. [1] (ii) Rotate triangle T through 90° clockwise about the point P. Label your image Y. [2] (b) Describe the single transformation which maps (i) flag P onto flag Q, Answer(b)(i) [3] (ii) flag P onto flag R. Answer(b)(ii) [2] 8 7 6 5 4 3 2 1 –1 –1 1 0 2 3 4 5 6 7 8 9 10 11 12 y x P Q R

12 © UCLES 2011 0581/31/M/J/11 For Examiner's Use 8 30 students took a vocabulary test. The marks they scored are shown below. 7 8 5 8 3 2 6 6 3 3 6 2 7 1 5 10 2 6 6 5 8 1 2 7 3 1 5 3 10 3 (a) Complete the frequency table below. The first five frequencies have been completed for you. You may use the tally column to help you. Mark Tally Frequency 1 3 2 4 3 6 4 0 5 4 6 7 8 9 10 [3]

13 © UCLES 2011 0581/31/M/J/11 [Turn over For Examiner's Use (b) (i) Find the range. Answer(b)(i) [1] (ii) Write down the mode. Answer(b)(ii) [1] (iii) Find the median. Answer(b)(iii) [2] (iv) Calculate the mean. Answer(b)(iv) [3] (c) A student is chosen at random. Find the probability that the student scored (i) 1 mark, Answer(c)(i) [1] (ii) 4 marks, Answer(c)(ii) [1] (iii) fewer than 6 marks. Answer(c)(iii) [1]

14 © UCLES 2011 0581/31/M/J/11 For Examiner's Use 9 (a) In the space below, construct the triangle ABC with AB = 10 cm and AC = 12 cm. Leave in your construction arcs. The line BC is already drawn. B C [2]

15 © UCLES 2011 0581/31/M/J/11 [Turn over For Examiner's Use (b) Measure angle ABC. Answer(b) Angle ABC = [1] (c) (i) Using a straight edge and compasses only, and leaving in your construction arcs, construct the perpendicular bisector of BC. [2] (ii) This bisector cuts AC at P. Mark the position of P on the diagram and measure AP. Answer(c)(ii) AP = cm [1] (d) Construct the locus of all the points inside the triangle which are 5 cm from A. [1] (e) Shade the region inside the triangle which is • nearer to B than to C and • less than 5 cm from A. [2] Question 10 is printed on the next page.

16 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 0581/31/M/J/11 For Examiner's Use 10 (a) B is 120 m from A on a bearing of 053°. Calculate (i) the distance d, Answer(a)(i) d = m [2] (ii) the bearing of A from B. Answer(a)(ii) [1] (b) 20 m 9 m 24 m G H A F NOT TO SCALE A vertical flagpole, AF, is 9 m high. It is held in place by two straight wires FG and FH. FG = 20 m and AH = 24 m. G, A and H lie in a straight line on horizontal ground. Calculate (i) angle FHA, Answer(b)(i) Angle FHA = [2] (ii) the distance GA. Answer(b)(ii) GA = m [3] 53° A B 120 m North North d NOT TO SCALE

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2011 question paper for the guidance of teachers 0581 MATHEMATICS 0581/31 Paper 3 (Core), maximum raw mark 104 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 May/June 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 – May/June 2011 0581 31 © 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 Mark 1 (a) (b) (c) (d) (e) 342.63 280 71.4 or 71.42 to 71.43 4.12 correct working 2 3 1ft 2 1 M1 for 500 ÷ 1.4593 M1 for 2 × 62 + 3 × 52 B1 for 124 or 156 seen B1 for 6 × 0.98 seen B1 for 5.88 or 4 + 6 × 0.02 50 × 2.54 = 127 oe or 130 ÷ 2.54 = 51.2 or better 2 (a) (b) (c) (i) (ii) (d) (triangular) prism 49.6 to 50.4 6 42 3.5 1 1 2 2ft 2ft M1 for ½ × 4 × 3 oe M1 for their (c)(i) × 7 M1 for their (c)(ii) ÷ (3 × 4) oe 3 (a) (i) (ii) (b) (c) (d) (e) 10 8 7fg – g3 6h(3h – 2j) 8 15 − t 9 2 3 2 2 2 3 M1 3 × 2 − −4 or better M1 for 19 = 3m − 5 oe M1 for m = (19 + 5) ÷ 3 oe B1 for 7fg or B1 for –g3 B1 for partial factorisation 2(9h2 – 6hj) or 3(6h2 – 4hj) or h(18h – 12j) or 6(3h2 – 2hj) or 3h(6h – 4j) or 2h(9h – 6j) or B1 for 6h(ah – 2j) or 6h(3h – bj) M1 for correct first step or M1 for correct second step ft M1 for 3p − 15 M1 for collecting their terms 2p = k or kp = 18

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0581 31 © University of Cambridge International Examinations 2011 4 (a) (i) (ii) (iii) (b) (i) (ii) (iii) (c) 1 15 10 3 24 6.67 or 6.66(6...) horizontal line to (105, 5) line from (their 105, 5) to (10 + their 105, 0) 1 1 1 1 2 3 1 1ft M1 for 4 ÷ 10 × 60 or M1 for 4 ÷ 6 1 , 4 × 6, (4 × 60)/10 oe M1 for dist = 5 and time = 45 seen M1 for 5 ÷ 45 × 60 oe 5 (a) (i) (ii) (b) (i) (ii) (iii) (c) 2 2x + 1 2 −2 2 7 points correct smooth curve −1.5 to −1.3 cao 1.3 to 1.5 cao (−1, −1) and (3, 7) cao 2 2ft 2 3 ft 1 1 1 1, 1 M1 for numbers representing change in y / change in x Implied by 2k/k M1 for {their (a)(i)}x + j or kx + 1 (j,k not equal to 0) B1 for 2 correct B2 for 5 or 6 points correct B1 for 3 or 4 points correct Must be close to parabolic in shape

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0581 31 © University of Cambridge International Examinations 2011 6 (a) (i) (ii) (iii) (iv) (b) (c) (d) 144 125 103 159 23 × 11 or 2 × 2 × 2 × 11 24 60 1 1 1 1 2 2 2 SC1 for 2 and 11 seen, no extras or SC1 for 2 × 4 × 11 SC1 for at least one of 2, 3, 4, 6, 8 or 12 or SC1 for 72 = 3 × 24 and 96 = 4 × 24 SC1 for 60k or SC1 2×2×3×5 oe 7 (a) (i) (ii) (b) (i) (ii) correct reflection correct rotation enlargement sf 2 about origin translation by       5 3 1 2 1 1 1 1 1 SC1 for rotation 90° anti-clockwise or 90° clockwise about any other point independent marks independent marks 8 (a) (b) (i) (ii) (iii) (iv) (c) (i) (ii) (iii) frequencies 5, 3, 3, 0, 2 9 3 5 4.8 30 3 oe 0 30 17 oe 3 1 1ft 2 3 1 1 1 B2 for 4 correct, B1 for 3 correct If frequencies blank then SC2 for all tallies correct, SC1 for 3 M1 clear attempt to find middle M1 for Σ their f × x implied by 144 – clear attempt M1 dep for dividing by 30 isw allow 0/30 only, accept zero, none, impossible accept 0.566 to 0.567 isw

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0581 31 © University of Cambridge International Examinations 2011 9 (a) (b) (c) (i) (ii) (d) (e) correct triangle with arcs 68° to 71° perpendicular bisector with 2 pairs of arcs 3 to 3.4 cm arc centre their A radius 5 cm shading inside arc and to left of perpendicular bisector 2 1ft 2 1ft 1ft 2 B1 without arcs or SC1 correct mirror image with arcs SC1 if accurate without arcs or accurate arcs with no line or accurate with arcs of AB or AC for their P on their bisector minimum must cut their AB and AC SC1 for either condition met 10 (a) (i) (ii) (b) (i) (ii) 95.8 or 95.83 to 95.84 233° 20.6° or 20.55 to 20.56 17.9 2 1cao 2 3 M1 for 120 × sin 53 or sin 53 =120 x oe M1 for tan = 24 9 oe M2 for 2 2 9 20 − or M1 for x2 + 92 = 202 oe