Cambridge IGCSE Mathematics (with coursework) 0581 — 2010 May/June Paper 3 · Variant 1

This document consists of 12 printed pages. IB10 06_0581_31/FP © UCLES 2010 [Turn over *7294417451* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/31 Paper 3 (Core) May/June 2010 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 2010 0581/31/M/J/10 For Examiner's Use 1 The population of a village is 2250. (a) 32% of the population are children. Calculate the number of children in the village. Answer(a) [2] (b) 360 people in the village are over the age of 60. (i) For these 360 people, the ratio of men to women is 2 : 7. Calculate how many men are over the age of 60. Answer(b)(i) [2] (ii) Write 360 as a fraction of 2250 in its lowest terms. Answer(b)(ii) [2] (c) The population of 2250 is expected to increase by 18% next year. Calculate the expected population next year. Answer(c) [3] (d) Write the number 2250 in standard form. Answer(d) [1] (e) Another village has a population of 1770, correct to the nearest ten. Write down the lower bound for the population of this village. Answer(e) [1]

3 © UCLES 2010 0581/31/M/J/10 [Turn over For Examiner's Use 2 The diagram shows a block of stone in the shape of a prism of length 42 cm. The cross-section is a trapezium ABCD. AB = 19 cm, AD = 10 cm, DC = 13 cm and angle ADC = 90°. (a) Calculate (i) the perimeter of the rectangular face ABFE, Answer(a)(i) cm [2] (ii) the area of the cross-section ABCD, Answer(a)(ii) cm2 [3] (iii) the volume of the block of stone. Answer(a)(iii) cm3 [2] (b) The mass of 1 cubic centimetre of the stone is 4 grams. Calculate the mass of the block. Give your answer in kilograms. Answer(b) kg [3] A B D C E H G F 13 cm 19 cm 42 cm 10 cm NOT TO SCALE

4 © UCLES 2010 0581/31/M/J/10 For Examiner's Use 3 Twelve students each answer 30 questions in a quiz. The time taken and the number of correct answers for each student is given in the table. Time taken in minutes 9 4 5 10 3 2 8 8 4 5 6 7 Number of correct answers 19 28 26 17 30 26 25 20 23 21 24 22 (a) Complete the scatter diagram below to show this information. The first six points have been plotted for you. 31 30 29 28 27 26 25 24 23 22 21 20 19 18 17 16 15 0 1 2 3 4 5 6 Time taken in minutes 7 8 9 10 11 Number of correct answers [3]

5 © UCLES 2010 0581/31/M/J/10 [Turn over For Examiner's Use (b) What type of correlation does the scatter diagram show? Answer(b) [1] (c) (i) Find the range of the time taken. Answer(c)(i) min [1] (ii) Calculate the mean time taken. Answer(c)(ii) min [3] (d) (i) Find the mode for the number of correct answers. Answer(d)(i) [1] (ii) Find the median for the number of correct answers. Answer(d)(ii) [1] (e) One of the 12 students is selected at random. Write down the probability that the student (i) took more than 8 minutes to answer the quiz, Answer(e)(i) [1] (ii) took less than 5 minutes and had more than 24 correct answers. Answer(e)(ii) [2]

6 © UCLES 2010 0581/31/M/J/10 For Examiner's Use 4 A N B C 29° 8.6 cm 13.4 cm NOT TO SCALE In triangle ABC, AN = 8.6 cm and is perpendicular to BC. Angle BAN = 29° and AC = 13.4 cm. (a) Use trigonometry to calculate (i) the length of BN, Answer(a)(i) BN = cm [3] (ii) angle CAN. Answer(a)(ii) Angle CAN = [2] (b) Calculate the length of NC. Answer(b) NC = cm [3]

7 © UCLES 2010 0581/31/M/J/10 [Turn over For Examiner's Use 5 7 6 5 4 3 2 1 –1 –2 –3 –4 0 –1 –2 –3 –4 –5 –6 –7 5 6 7 4 3 2 1 y x A B C D W V U T R Q P S (a) On the grid, draw the image of (i) the flag ABCD after translation by       −3 4 , [2] (ii) the flag ABCD after enlargement, scale factor 2, centre the origin, [2] (iii) the flag ABCD after reflection in the x-axis. [2] (b) Describe fully the single transformation which maps ABCD onto PQRS. [2] (c) Describe fully the single transformation which maps ABCD onto TUVW. [3]

8 © UCLES 2010 0581/31/M/J/10 For Examiner's Use 6 (a) Complete the table of values for the function x y 3 = , x ≠ 0. x −3 −2.5 −2 −1.5 −1 −0.5 −0.3 0.3 0.5 1 1.5 2 2.5 3 y −1 −1.2 −2 −3 −6 3 2 1.5 1 [3] (b) On the grid below, draw the graph of x y 3 = for −3 Y x Y −0.3 and 0.3 Y x Y 3. [5] y x 10 9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 0 –1 1 2 3 –2 –3

9 © UCLES 2010 0581/31/M/J/10 [Turn over For Examiner's Use (c) Use your graph to solve the equation x 3 = 7. Answer(c) x = [1] (d) Complete the table of values for 1 3 2 − = x y . x −3 0 3 y [2] (e) On the grid, draw the straight line 1 3 2 − = x y for −3 Y x Y 3. [2] (f) Write down the co-ordinates of the points where the line 1 3 2 − = x y intersects the graph of x y 3 = . Answer(f) ( , ) and ( , ) [2] 7 4 S a d = + (a) Find S when a = 17 and d = − 5. Answer(a) S = [2] (b) Find d when S = 37 and a = 5. Answer(b) d = [2] (c) Make d the subject of the formula 4 S a d = + . Answer(c) d = [2]

10 © UCLES 2010 0581/31/M/J/10 For Examiner's Use 8 In this question give all your answers to 2 decimal places. (a) Ankuri lends her brother $275 for 4 years at a rate of 3.6% per year simple interest. Calculate the total amount her brother owes after 4 years. Answer(a) $ [3] (b) Monesh invests $650 in a bank which pays 4% per year compound interest. Calculate the amount Monesh will have after 2 years. Answer(b) $ [3] (c) Theresa and Ian have 400 euros (€) each. (i) Theresa changes her €400 for pounds (£) when the exchange rate is €1= £ 0.7857. Calculate the amount she receives. Answer(c)(i) £ [2] (ii) Ian changes his €400 for dollars ($) when the exchange rate is $1= € 0.6374. Calculate the amount he receives. Answer(c)(ii) $ [3]

11 © UCLES 2010 0581/31/M/J/10 [Turn over For Examiner's Use 9 Triangle ABC is drawn accurately. (a) Measure and write down (i) the length of AC, Answer(a)(i) AC = cm [1] (ii) the size of angle CAB. Answer(a)(ii) Angle CAB = [1] (b) Construct accurately the locus of all the points 7 cm from C. [2] (c) The point X lies outside the triangle ABC, with CX = 7 cm and angle BCX= 67°. Draw accurately the line CX. [2] (d) Draw the line BX. Measure and write down the length of this line. Answer(d) BX = cm [1] (e) Using a straight edge and compasses only, construct the locus of points equidistant from BC and from BX. [2] Question 10 is printed on the next page. C A B

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 0581/31/M/J/10 For Examiner's Use 10 Look at the sequence of diagrams. (a) Diagram 2 has a height of 2. Write down the height of (i) Diagram 5, Answer(a)(i) [1] (ii) Diagram 10, Answer(a)(ii) [1] (iii) Diagram n. Answer(a)(iii) [1] (b) Diagram 2 has a width of 3. Find the width of (i) Diagram 5, Answer(b)(i) [1] (ii) Diagram 10, Answer(b)(ii) [1] (iii) Diagram n. Answer(b)(iii) [2] (c) There are 6 squares in Diagram 2 and 15 squares in Diagram 3. (i) Write down how many squares there are in Diagram 5. Answer(c)(i) [1] (ii) Explain how this is found from the height and width of the diagram. Answer(c)(ii) [1] (iii) Write down, in terms of n, how many squares there are in Diagram n. Answer(c)(iii) [1] Diagram 1 Diagram 2 Diagram 3 Diagram 4

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 0581/31 Paper 31 (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. • 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 31 © UCLES 2010 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 art anything rounding to soi seen or implied Qu. Answers Mark Part Marks 1 (a) 720 2 M1 100 2250 32× (b) (i) 80 2 M1 360 7 2 2 × + (ii) 25 4 2 W1 for 180/1125, 120/750, 72/450, 60/375, 36/225, 24/150, 12/75, 20/125, 8/50 (c) 2655 3 M2 2250 100 118 × oe If M0 then M1 for 2250 100 18 × or 405 seen (d) 2.25 × 103 cao 1 (e) 1765 cao 1 2 (a) (i) 122 2 M1 for 2 × 19 + 2 × 42 oe (ii) 160 3 M2 for ( ) 10 13 19 2 1 × + oe SC1 for rectangle 130 or triangle 30, 65, 95 (iii) 6720 or their (a)(ii) × 42 evaluated 2ft M1 their (a)(ii) × 42 (b) 26.88 or their (a)(iii) × 0.004 evaluated or 26.9 3ft M1 their (a)(iii) × 4 soi M1 division by 1000 soi

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0581 31 © UCLES 2010 3 (a) (b) (c) (i) (ii) (d) (i) (ii) 6 points correctly plotted negative cao 8 cao art 5.92 26 cao 23.5 cao 3 1 1 3 1 1 P2 for 4 or 5 points, P1 for 2 or 3 points M1 for attempt to add the 12 values (for time) implied by 71 M1 dep for division by 12 SC1 for 23.4 (e) (i) 12 2 oe 1 0.166 or 0.167 or 16.6% or 16.7% (ii) 12 3 oe 2 0.25 or 25% SC1 for (4,28) (2,26) (3,30) listed or ringed on diagram or table 4 (a) (i) art 4.77 3 M2 for BN = 8.6 × tan 29 oe or M1 for 29 tan 6.8 = BN oe (ii) art 50.1° 2 M1 for cos CAN = 8.6 ÷ 13.4 (b) 10.2 to 10.3 3 M1 for 13.42 – 8.62 (105.6) M1 dep for 2 2 6.8 4. 13 − 5 (a) (i) correct image 2 B1 for translation by     k 4 or     −3 k or    − 4 3 (ii) (iii) (b) (c) correct image correct image Reflection, x = −2 Rotation, origin, 90° (anti-clockwise or +90°) 2 2 2 3 B1 for figure of correct size and orientation in wrong position B1 for reflection in y-axis or in any horizontal line. B1 each B1 each accept 270° clockwise, −270°,1/4

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0581 31 © UCLES 2010 6 (a) (b) (c) (d) (e) (f) –1.5 –10 10 6 1.2 14 points plotted accurately 2 smooth correct curves No part across y-axis 0.4 to 0.5 −3 −1 1 Ruled line from (−3, −3) to (3, 1) (−1.5, −2) and (3, 1) 3 P3ft C1 B1 1 2 2 1, 1 B2 for 3 or 4 correct, B1 for 2 correct P2ft for 11, 12 or 13 points, P1ft for 8, 9 or 10 Indep B1 for 2 correct SC1 for freehand or short ruled line – must meet curve twice or P1 for their 3 points plotted 7 (a) (b) −3 8 2 2 1 for correct substitution seen M1 for 37−5 =4d oe (c) 4 a S − 2 M1 for one correct step seen 8 (a) 314.60 3 M1 for 100 6.3 4 275 × × or 39.6 M1 dep for their interest added to 275 (b) 703.04 3 M2 for 650 × 1.042 or M1 for 650 × 1.04 oe (implied by 676) and M1 dep for second year (c) (i) (ii) 314.28 627.55 or 627.54 2 3 M1 for 400 × 0.7857 M1 for 400 ÷ 0.6374 soi A1 627.54…, 628, 627.5 B1 indep for their visible answer corrected to 2dp Penalise accuracy only once in the question 9 (a) (i) (ii) (b) (c) (d) (e) 9 or 8.9 to 9.1 53 – 55 compass drawn circle centre C radius 7 cm correct line drawn with angle BCX = 67° in range 9.3 – 9.9 ruled accurate angle bisector of their CBX with 2 pairs of arcs 1 1 2 2ft 1ft 2ft SC1 incomplete accurate circle SC1 any complete circle centre C SC1 for BCX = 113° or BCX = 67° inside triangle or BCX = 67°, CX not = 7 Strict ft from (c) SC1 if accurate but without arcs or M1 for 2 pairs of arcs

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2010 0581 31 © UCLES 2010 10 (a) (i) (ii) (iii) (b) (i) (ii) (iii) (c) (i) (ii) (iii) 5 10 n 9 19 2n − 1 oe 45 5 × 9 n(2n – 1) oe or n(their (b)(iii)) 1 1 1 1 1 2 1 1 1ft SC1 for 2n + k oe or jn – 1, j not = 0 Accept height × width Their (a)(iii) × their (b)(iii)