Cambridge IGCSE Mathematics - International 0607 — 2009 May/June Paper 3 · Variant 1

This document consists of 16 printed pages. IB09 06_0607_03/3RP © UCLES 2009 [Turn over *5760555781* For Examiner's Use UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/03 Paper 3 (Core) May/June 2009 1 hour 45 minutes Candidates answer on the Question Paper Additional Materials: Geometrical Instruments 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 all the questions. Unless instructed otherwise, give your answers exactly or correct to three significant figures as appropriate. Answers in degrees should be given to one decimal place. For π, use your calculator value. You must show all the relevant working to gain full marks and you will be given marks for correct methods, including sketches, even if your answer is incorrect. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of the marks for this paper is 96. www.XtremePapers.com

2 © UCLES 2009 0607/03/M/J/09 Formula List Area, A, of triangle, base b, height h. A = 1 2 bh Area, A, of circle, radius r. A = πr2 Circumference, C, of circle, radius r. C = 2πr Curved surface area, A, of cylinder of radius r, height h. A = 2πrh Curved surface area, A, of cone of radius r, sloping edge l. A = πrl Curved surface area, A, of sphere of radius r. A = 4πr2 Volume, V, of prism, cross-sectional area A, length l. V =Al Volume, V, of pyramid, base area A, height h. V= 1 3 Ah Volume, V, of cylinder of radius r, height h. V = πr2h Volume, V, of cone of radius r, height h. V = 1 3 πr2h Volume, V, of sphere of radius r. V = 4 3 πr3

3 © UCLES 2009 0607/03/M/J/09 [Turn over For Examiner's Use Answer all the questions. 1 55 7 27 W C U 11 100 students are asked if they walk (W) or cycle (C) as part of their regular exercise. The Venn diagram shows this information. (a) How many students (i) walk and cycle, Answer (a)(i) [1] (ii) cycle but do not walk, Answer (a)(ii) [1] (iii) do not walk and do not cycle? Answer (a)(iii) [1] (b) Write down the value of (i) n(W), Answer (b)(i) [1] (ii) n(C ′ ). Answer (b)(ii) [1] (c) One of the students is chosen at random. Find the probability that this student does at least one of these types of exercise. Answer (c) [1] (d) A school has 2000 students. Use your results to predict the number of students from the school who do at least one of these types of exercise. Answer (d) [1]

4 © UCLES 2009 0607/03/M/J/09 For Examiner's Use 2 Konrad keeps a record of the marks he receives in ten tests. Mark 7 8 9 10 Frequency 5 2 2 1 (a) Find (i) the mode, Answer (a)(i) [1] (ii) the median, Answer (a)(ii) [1] (iii) the mean, Answer (a)(iii) [1] (iv) the range, Answer (a)(iv) [1] (v) the upper quartile. Answer (a)(v) [1] (b) A pie chart to show this information has been started below. Complete and label the pie chart accurately. 7 marks 8 marks [2]

5 © UCLES 2009 0607/03/M/J/09 [Turn over For Examiner's Use (c) 6 5 4 3 2 1 0 8 7 10 9 Frequency Mark On the grid above, draw a bar chart to show Konrad’s marks. [3] (d) Work out the percentage of the ten tests in which Konrad’s marks were 9 or 10. Answer (d) % [2]

6 © UCLES 2009 0607/03/M/J/09 For Examiner's Use 3 (a) The ratio of Abdul’s height to Babar’s height is Abdul : Babar = 6 : 5 Abdul’s height is 180 cm. Calculate Babar’s height. Answer (a) cm [2] (b) The masses of Abdul and Babar are in the same ratio as their heights. The total of their masses is 121 kg. Show that Abdul’s mass is 66 kg. [2] (c) Last year Abdul’s mass was 63 kg. It is now 66 kg. Calculate the percentage increase in Abdul’s mass. Answer (c) % [2]

7 © UCLES 2009 0607/03/M/J/09 [Turn over For Examiner's Use (d) Convert 63 kg into grams. Give your answer in standard form. Answer (d) g [3] (e) Abdul and Babar run 100 metres. (i) Abdul runs at a steady speed of 7 metres per second. Find the time taken for Abdul to run the 100 metres. Answer (e)(i) seconds [2] (ii) Babar takes 14.5 seconds to run the 100 metres. Find his speed, in metres per second. Answer (e)(ii) m/s [2]

8 © UCLES 2009 0607/03/M/J/09 For Examiner's Use 4 0 –2 2 3 –1 y x The diagram shows a sketch of the graph of the function y = 3 – x2. (a) On the diagram, sketch the graph of the function y = 2 x + 2 for −2 Y x Y=2. [2] (b) Solve the equation 3 − x2 = 2 x + 2. Give your answers correct to 4 decimal places. Answer (b) x = or [2] (c) On the diagram, sketch the straight line y = 4. From your diagram, explain why the equation 3 – x2 = 4 has no solutions. [1]

9 © UCLES 2009 0607/03/M/J/09 [Turn over For Examiner's Use 5 75° 35° 6 cm 12 cm D C B A 7 cm NOT TO SCALE ABCD is a trapezium with AB parallel to DC. AB = 12 cm, DC = 6 cm and the height of the trapezium is 7 cm. Angle DAB = 75° and angle CDB = 35°. Calculate (a) the area of triangle ABD, Answer (a) cm2 [2] (b) the area of the trapezium, Answer (b) cm2 [2] (c) angle ADC, Answer (c) [1] (d) angle ABD. Answer (d) [1]

10 © UCLES 2009 0607/03/M/J/09 For Examiner's Use 6 Each day Lavinia records the number of glasses of water and the number of cups of coffee she drinks. The results for one week are shown in the table. Day Sun Mon Tue Wed Thu Fri Sat Number of glasses of water 8 5 6 3 7 7 6 Number of cups of coffee 2 4 4 6 2 1 2 (a) On the grid, draw a scatter diagram to show this information. 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 Number of glasses of water Number of cups of coffee [3] (b) Which of these words (positive, negative, none) describes the correlation between the number of glasses of water and the number of cups of coffee? Answer (b) [1] (c) (i) Calculate the mean number of cups of coffee. Answer (c)(i) [1] (ii) The mean number of glasses of water is 6. Draw the line of best fit for this data. [2]

11 © UCLES 2009 0607/03/M/J/09 [Turn over For Examiner's Use 7 4 cm NOT TO SCALE 3 cm 5 cm The diagram shows a solid cone of height 4 cm and radius 3cm. The length of the sloping edge of the cone is 5 cm. (a) (i) Calculate the volume of the cone. Answer (a) (i) cm3 [2] (ii) The cone is made of metal. 1 cm3 of the metal has a mass of 7.5 g. Calculate the mass of the cone. Answer (a) (ii) g [2] (b) (i) Calculate the total surface area of the cone. Answer (b) (i) cm2 [3] (ii) Change your answer into square metres. Answer (b) (ii) m2 [1] (iii) One pot of paint covers 7 m2. How many of these cones can be painted using one pot of paint? Answer (b) (iii) [2]

12 © UCLES 2009 0607/03/M/J/09 For Examiner's Use 8 7 –6 –2 4 y x 0 (a) On the diagram, sketch the graph of the function y = 3 3 x − x2 + 1 for −2 Y x Y=4. [4] (b) Find the value of y when (i) x = 0 Answer (b)(i) [1] (ii) x = 3.5 Answer (b)(ii) [1] (c) Find the three values of x when y = 0. Answer (c) x = , , [3] (d) Find the co-ordinates of the local minimum point. Answer (d) ( , ) [2] (e) Find the value of x when 3 3 x − x2 + 1 = −2 Answer (e) [1] (f) The domain of the function f(x) = 3 3 x − x2 + 1 is −2 Y x Y=4. Find the range of the function. Answer (f) [2]

13 © UCLES 2009 0607/03/M/J/09 [Turn over For Examiner's Use 9 O P Q T 70° NOT TO SCALE The diagram shows a circle, centre O. TP and TQ are tangents to the circle at P and Q. Angle PTQ = 70°. (a) Calculate (i) angle TPQ, Answer (a)(i) [2] (ii) angle POQ. Answer (a)(ii) [2] (b) Another circle can be drawn through the points O, P, T and Q. Complete the statement. “OT is a of this new circle.” [1]

14 © UCLES 2009 0607/03/M/J/09 For Examiner's Use 10 0 L (1,2) M (1,6) K (8,2) y x NOT TO SCALE The diagram shows triangle KLM on a co-ordinate grid. The diagram is not to scale so do not measure any lengths or angles. (a) Write down the equation of the straight line which passes through L and M. Answer (a) [1] (b) Find the gradient of the line MK. Answer (b) [2]

15 © UCLES 2009 0607/03/M/J/09 [Turn over For Examiner's Use (c) Find the co-ordinates of the midpoint of the line MK. Answer (c) ( , ) [2] (d) Calculate the length of MK. Answer (d) [2] (e) Use trigonometry to calculate the size of angle LKM. Answer (e) [2] Question 11 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. 0607/03/M/J/09 For Examiner's Use 11 (a) Find the value of 2 _ 5 2 x y when x = 7 and y = 4. Give your answer as a fraction in its lowest terms. Answer (a) [2] (b) 2 5 2 x y − can be written as a single fraction 10 x y − . Fill in the two missing values. [2] (c) 2 5 2 x y − = 1 and y = 14. Find the value of x. Answer (c) x = [2] (d) 2 5 2 x y − = 1. Find y in terms of x. Answer (d) y = [2]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2009 question paper for the guidance of teachers 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/03 Paper 3 (Core), maximum raw mark 96 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 2009 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 2009 0607 03 © UCLES 2009 M marks are given for a correct method. A marks are given for an accurate answer following a correct method. B marks are given for a correct statement or step. D marks are given for a clear and appropriately accurate drawing. P marks are given for accurate plotting of points. E marks are given for correctly explaining or establishing a given result. Abbreviations cao correct answer only cso correct solution only ft follow through oe or equivalent soi seen or implied ww without working www without wrong working 1 (a) (i) 55 B1 (ii) 7 B1 (iii) 11 B1 (b) (i) 82 B1 (ii) 38 B1 (c) 100 89 oe B1 (d) 1780 ft B1 ft their (c) × 2000 [7] 2 (a) (i) 7 B1 (ii) 7.5 B1 (iii) 7.9 B1 (iv) 3 B1 (v) 9 B1 (b) Radius drawn giving angles of 72° and 36° ±2° Labels 9 and 10 correctly placed B1 B1 Must be ruled If 2 sectors and 9 is larger (c) Ruled bars of heights 5, 2, 2, 1 B3 B2 for 3 correct, B1 for 2 correct. Deduct 1 for freehand but reasonable (d) 30 B2 If B0, M1 for 3 ÷ 10 × 100 oe [12]

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 03 © UCLES 2009 3 (a) 150 B2 If B0, M1 for180 ÷ 6 × 5 (b) 121 ÷ 11 oe × 6 oe M1 M1 Accept 6 × 11 = 66, 5 × 11 = 55 M1 55 + 66 = 121 M1 (c) 4.76 (4.761 – 4.762)) B2 If B0, M1 for 3 ÷ 63 × 100 oe (d) 63 × 1000 6.3 × 104 www3 M1 A1A1 SC2 for 63 × 103 oe (e) (i) 14.3 (14.28 – 14.29) B2 M1 for 100 ÷ 7 (ii) 6.9(0) (6.896 – 6.897) B2 If B0, M1 for 100 ÷ 14.5 [13] 4 (a) 2 2 + = x y drawn B1 B1 For approx straight line with grad approx 2 1 , reaching curve twice For approx straight line with y-intercept approx 2, also reaching curve twice (b) – 1.2808, 0.7808 B1B1 Accept 2dp (–1.28 and 0.78) or better – 1.281 to –1.280 , 0.781 or 0.7807 to 0.7808 SC1 – 1.3, 0.8 (c) Line would not meet the curve even if extended oe B1 [5] 5 (a) 42 B2 If B0, M1 for 0.5 × 12 × 7 (b) 63 B2 If B0, M1 for 0.5 × 6 × 7 + their 42 oe (c) 105 B1 (d) 35 B1 [6]

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 03 © UCLES 2009 6 (a) 7 correct points P3 P2 for 5 or 6 correct, P1 for 3 or 4 (b) Negative B1 (c) (i) 3 B1 (ii) Straight line through (6, their 3) with negative gradient through (8, 0.5 to 1.5) M1 A1 Plotted or implied (within 2 mm of ( y x, )) A0 if not ruled For 3 to 8 at least [7] 7 (a) (i) 37.68 – 37.7 B2 If B0, M1 for 3 1 × π × 32 × 4 Accept 12π (ii) 283 (282.6 – 282.8) ft B2 ft If B0, M1 for (i) × 7.5 (b) (i) 75.36 – 75.41 B3 If B0, M1 for π × 3 × 5 M1 for π × 32 Accept 24π (ii) 0.007536 – 0.007541 ft B1 ft ft their (i) (iii) 928 cao B2 If B0, M1 for 7 ÷ their (ii) [10] 8 (a) Cubic shape with max then min Cross x-axis 3 times 2 x-intercepts positive Max point close to (0, 1) B1 B1 B1 B1 At least from –2 to 4 Dependent on previous B1 Dependent on first B1 (b) (i) 1 B1 (ii) 3.04(3.041 – 3.042) B1 (c) –0.879, 1.35, 2.53 (–0.8794 to –0.8793, 1.347…, 2.532...) B1,B1, B1 If B0, SC2 for –0.88, 1.3, 2.5 or SC1 for 2 of these. If B1, SC1 for other two to 2 sf (d) (2, –0.333) B1,B1 Allow –0.33 or 3 1 − (e) –1.43 (–1.426 to –1.425) B1 (f) –5.67 to 6.33 (–5.666 to – 5.667 to 6.3333.. ) oe B1, B1 Allow –5.6 or –5.7 and 6.3 [14]

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2009 0607 03 © UCLES 2009 9 (a) (i) 55 B2 M1 for 2 1 (180 – 70) (ii) 110 B2 M1 for 180–70 or for 360–90–90–70 soi (may be on diagram) or for 180 – 2[90 – their (i)] ft (b) Diameter B1 [5] 10 (a) x = 1 B1 (b) 7 4 − oe –0.571 or –0.5714… B2 B1 for – ve, B1 for 4/7 Allow –0.57 (c) (4.5, 4) B1, B1 (d) (their 4)2 + (their 7)2 8.06 (8.062…) ft www2 M1ft A1ft ft from (b) (e) tan(angle) = 7 4 their their oe 29.7 (29.74 – 29.75….) ft www2 M1ft A1ft ft from (b) or (d) Radians 0.519 give M1A1 [9] 11 (a) 5 4 cao B2 If B0, M1 for 2 4 5 7 2 − × implied by 10 8 oe (b) 4, 5 B1,B1 May be embedded (c) 20 B2 If B0, M1 for 8 5 2 = x (Must reach correct equation with one variable term and one constant term only.) (d) 1 5 2 2 − = x y oe       − = 1 5 2 2 x y oe www 2 2 5 4 − x , 5 10 4 − x M1 M1 for re-arranging correctly for + or – 2 y oe for multiplying by 2 correctly (any order) Mark final answer [8]