Cambridge IGCSE Mathematics - International 0607 — 2011 May/June Paper 3 · Variant 1

This document consists of 16 printed pages. IB11 06_0607_31/2RP © UCLES 2011 [Turn over *9835578859* For Examiner's Use UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/31 Paper 3 (Core) May/June 2011 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 marks for this paper is 96. www.XtremePapers.com

2 © UCLES 2011 0607/31/M/J/11 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 2011 0607/31/M/J/11 [Turn over For Examiner's Use Answer all the questions. 1 Ali and Amanda are in the same class at school. (a) In a test Ali’s mark is 24 and Amanda’s mark is 28. (i) Write down the ratio. Ali’s mark : Amanda’s mark. Give your answer in its simplest form. Answer(a)(i) : [1] (ii) Calculate Amanda’s mark as a percentage of Ali’s mark. Answer(a)(ii) % [2] (b) In another test Ali’s mark is again 24 but the ratio of the marks changes to Ali’s mark : Amanda’s mark = 8 : 7. Calculate Amanda’s mark. Answer(b) [2] (c) Ali and Amanda share $35 in the ratio 3 : 4. Calculate how much Ali receives. Answer(c) $ [2]

4 © UCLES 2011 0607/31/M/J/11 For Examiner's Use 2 (a) Simplify fully. (i) 4 3 12 4 x x × Answer(a)(i) [2] (ii) 3 15 15 3 x x ÷ Answer(a)(ii) [2] (iii) 2 6 3 x y y t × Answer(a)(iii) [2] (b) Write 2 5 2 c d + as a single fraction. Answer(b) [2] 3 A ferry leaves Calais at 23 15. It takes 1 h 55 min to reach Dover. (a) Write down the arrival time of the ferry at Dover. Answer(a) [1] (b) The distance travelled is 43 km. Calculate the average speed of the journey, in km/h. Answer(b) km/h [3] (c) In 2009 a ferry ticket cost €40. The cost of the ferry ticket increased each year by 5%. Calculate the cost of the ferry ticket in 2011. Answer(c) € [3]

5 © UCLES 2011 0607/31/M/J/11 [Turn over For Examiner's Use 4 5 4 3 2 1 –1 –2 –3 –4 –5 0 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 y x H W G (a) Describe fully the single transformation that maps triangle W onto (i) triangle G, [2] (ii) triangle H. [3] (b) On the grid, (i) draw the translation of triangle W by 1 3 −       , [2] (ii) draw the enlargement of triangle W, centre (0, 0), scale factor 1 2 . [2]

6 © UCLES 2011 0607/31/M/J/11 For Examiner's Use 5 (a) y = 3x – 8 (i) Find the value of y when x = –5. Answer(a)(i) [1] (ii) Make x the subject of the equation. Answer(a)(ii) x = [2] (b) Solve the simultaneous equations. Show your method. y = 2x – 7 y = 3 – 2x Answer(b) x = y = [3]

7 © UCLES 2011 0607/31/M/J/11 [Turn over For Examiner's Use 6 100 90 80 70 60 50 40 30 20 10 0 5 10 15 20 25 30 35 40 45 50 Cumulative frequency Test mark The cumulative frequency graph shows the distribution of test marks for 100 students. Use the graph to find (a) the median, Answer(a) [1] (b) the inter-quartile range, Answer(b) [2] (c) the number of students with a mark of at least 20. Answer(c) [2]

8 © UCLES 2011 0607/31/M/J/11 For Examiner's Use 7 P C B A D 10 cm 10 cm 12 cm NOT TO SCALE M The diagram shows a pyramid with a square horizontal base ABCD. The diagonals of the base intersect at M. The vertex, P, of the pyramid is vertically above M. AB = BC = 10 cm and PM = 12 cm. (a) Calculate the volume of the pyramid. Answer(a) cm3 [2]

9 © UCLES 2011 0607/31/M/J/11 [Turn over For Examiner's Use (b) P C B 10 cm 13 cm NOT TO SCALE The diagram shows one of the faces of the pyramid, triangle PBC. The distance from P to the midpoint of BC is 13 cm. Calculate (i) the area of triangle PBC, Answer(b)(i) cm2 [2] (ii) the total surface area of the pyramid. Answer(b)(ii) cm2 [2]

10 © UCLES 2011 0607/31/M/J/11 For Examiner's Use 8 32 students are asked how many coins they have. The results are shown in the pie chart. 45° 45° 4 coins 5 coins 0 coins 1 coin 2 coins 3 coins (a) (i) Measure the angle which shows the number of students who have 4 coins. Answer(a)(i) [1] (ii) Calculate the number of students who have 4 coins. Answer(a)(ii) [1] (iii) Calculate the number of students who have more than one coin. Answer(a)(iii) [2] (b) Complete the frequency table. Number of coins 0 1 2 3 4 5 Number of students (frequency) 2 6 [2] (c) Find (i) the mean, Answer(c)(i) [1] (ii) the mode, Answer(c)(ii) [1] (iii) the median. Answer(c)(iii) [1]

11 © UCLES 2011 0607/31/M/J/11 [Turn over For Examiner's Use 9 North North 140° 120 m P S Q NOT TO SCALE Q is 120 m from P, on a bearing of 140°. (a) Find the bearing of P from Q. Answer(a) [1] (b) S is due south of P and due west of Q. Calculate the distance SQ. Answer(b) m [3] (c) (i) R is also 120 m from P and is due west of S. Show R and the line PR on the diagram. [1] (ii) Find the bearing of R from P. Answer(c)(ii) [1]

12 © UCLES 2011 0607/31/M/J/11 For Examiner's Use 10 line 1 line 3 line 2 2 8 0 3 A 8 y x C B NOT TO SCALE The diagram shows three lines, line 1, line 2 and line 3. Line 1 is parallel to the y-axis and passes through (3, 0). Line 2 is parallel to the x-axis and passes through (0, 2). Line 3 passes through (8, 0) and (0, 8). (a) Find the equation of (i) line 1, Answer(a)(i) [1] (ii) line 2, Answer(a)(ii) [1] (iii) line 3. Answer(a)(iii) [2]

13 © UCLES 2011 0607/31/M/J/11 [Turn over For Examiner's Use (b) The lines intersect at the points A, B and C as shown in the diagram. (i) Work out the co-ordinates of B. Answer(b)(i) ( , ) [2] (ii) Work out the co-ordinates of the midpoint of AB. Answer(b)(ii) ( , ) [1] (iii) Calculate the length of BC. Answer(b)(iii) [3]

14 © UCLES 2011 0607/31/M/J/11 For Examiner's Use 11 AB is a diameter of a circle, centre O. T is a point on the circle and angle TAB = 40°. UTV is a tangent to the circle at T. (a) Complete the following statements. (i) Angle ATB = , because [1] (ii) Angle OTV = , because [1] (b) Find the size of (i) angle ATO, Answer(b)(i) [1] (ii) angle TOB, Answer(b)(ii) [1] (iii) angle UTB. Answer(b)(iii) [1] (c) AB and UV are extended to meet at X. (i) Show this on the diagram. [1] (ii) Calculate the size of angle TXO. Answer(c)(ii) [1] 40° A U V T O B NOT TO SCALE

15 © UCLES 2011 0607/31/M/J/11 [Turn over For Examiner's Use 12 y x 4 –4 –2 2 0 (a) On the axes, sketch the graph of (i) y = x2 – 2 for – 2 Y x Y 2, [2] (ii) y = 2x for – 2 Y x Y 2. [2] (b) Write down the zeros of y = x2 – 2. Answer(b) x = and x = [2] (c) Solve the equation 2x = x2 – 2 for – 2 Y x Y 2. Answer(c) x = [1] (d) For the domain – 2 Y x Y 2, write down the range of the function 2x. Answer(d) [2]

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 0607/31/M/J/11 For Examiner's Use 13 A bag contains 7 white beads and 4 black beads. Two beads are taken out of the bag at random (without replacement). (a) Complete the tree diagram by putting the probabilities in the spaces. First Bead Second Bead white black white black black white 7 11 6 10 [2] (b) Calculate the probability that (i) both beads are white, Answer(b)(i) [2] (ii) exactly one bead is white. Answer(b)(ii) [3]

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 0607 CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/31 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. • 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 0607 31 © University of Cambridge International Examinations 2011 1 (a) (i) 6 : 7 B1 (ii) 117 (116.6 to 116.7) ft B2 ft ft their (i) if used. If B0, M1 for 28 ÷ 24 (× 100) o.e. (b) 21 B2 If B0, M1 for 24 ÷ 8 × 7 or 3 or 168 seen (c) 15 B2 If B0, M1 for 35 ÷ 7 × 3 or 5 or 105 seen [7] 2 (a) (i) 48x7 B2 B1 for 48xk or kx7 (ii) 5x–12 or 12 5 x B2 B1 for 5xk or kx–12 or 12 x k or SC1 for 5–12 (iii) t x 4 final answer B2 M1 for ty xy 3 12 seen (or better) or correct cancelling of y and 3 seen (b) 10 5 4 d c + B2 M1 for 10 5 10 4 d c + or d c 5 4 + seen or common denominator of 10 [8] 3 (a) (0)1 10 B1 Accept any reasonable notation. (b) 22.39 to 22.44 B3 If B0, M1 for dist / time and M1 for converting minutes to hours M’s independent (Allow dividing by 1.55 for first M1) (c) 44.1(0) B3 M2 for 40 × 1.052 o.e. M1 for 40 × 1.05 o.e. (implied by 42) Answer of 44 implies M1 (i.e. first year) [7] 4 (a) (i) Reflection y = –1 B1 B1 Independent (ii) Rotation (0, 0) 90° (anti-clockwise) oe B1 B1 B1 Independent (b) (i) Triangle at (2, –2), (6, –2), (6, 0) B2 SC1 for translation       k 1 or       −3 k or       − 2 3 2 1 (ii) Triangle at (0.5, 0.5), (2.5, 0.5), (2.5, 1.5) B2 SC1 any other enlargement, sf 2 1 correct orientation or sf – 2 1 , centre (0, 0) [9]

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 31 © University of Cambridge International Examinations 2011 5 (a) (i) –23 B1 (ii) 3 8 + y oe www 2 M1 M1 Correctly re-arranging with x term isolated Correctly dividing M’s independent (b) 2.5, –2 M1 A1A1 M1 for correctly eliminating one variable to one equation in other, or for sketch of both lines, one positive gradient, one negative gradient and intersection in bottom right quadrant (can be freehand) trial and improvement both correct 3 (one correct 0) ww or other GDC applications both correct SC2 (one correct 0) [6] 6 (a) 27 B1 (b) 8 B2 B1 for (l.q. = ) 24 or (u.q. = ) 32 (c) 88 or 89 B2 M1 for 12 seen [5] 7 (a) 400 B2 If B0, M1 for 3 1 × 102 × 12 (b) (i) 65 B2 If B0, M1 for 0.5 × 10 × 13 (ii) 360 ft B2 ft ft their (i). If B0, M1 for 4 × their (i) + 102 [6] 8 (a) (i) 135° B1 2 ± ° (ii) 12 ft B1 ft ft their (a)(i) only if their angle gives an integer (iii) 24 B2 M1 for 360 90 or 360 270 or 8 (b) 4, 4, …., …., 12, 4 ft B2 ft B1 for 3 correct ft their (a)(ii) (c) (i) 2.9375 or 2.938 or 2.94 ft B1 ft ft their (b) (ii) 4 ft B1 ft ft their (b) (iii) 3.5 ft B1 ft ft their (b) [9] 9 (a) 320 B1 (b) 77.1 (3…..) B3 If B0, B1 for angle P = 40 (or Q = 50) (may be on diagram), M1 for sin40 = 120 SQ oe (c) (i) R shown on diagram to make triangle PQR look isosceles (may be freehand) B1 (ii) 220 B1 [6]

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 31 © University of Cambridge International Examinations 2011 10 (a) (i) x = 3 B1 (ii) y = 2 B1 SC1 if (i) is y = 3 and (ii) is x = 2 (iii) x + y = 8 oe B2 If B0, M1 for gradient = 8 8 − (or better) or x + y = k (b) (i) (6, 2) ft B1B1 ft ft their line 2, line 3 but can recover (ii) (4.5, 2) cao B1 (iii) 4.24 (4.242 to 4.243) ft www 3 B3 ft M2 for 32 + (their AB)2 (If M0, B1 for AC = 3) ft their x-coord of B for AB Accept 18 or 2 3 [10] 11 (a) (i) 90 and semi-circle B1 Allow AB is diameter as reason. Allow right angle for 90. (ii) 90 and tangent/radius B1 Allow right angle for 90. (b) (i) 40 B1 (ii) 80 B1 (iii) 140 B1 (c) (i) AB and UV extended to meet at X (may be freehand) B1 (ii) 10 B1 [7] 12 (a) B1 B1 B1 B1 B1 U-shaped parabola, cutting x-axis twice. B1 symmetry about y-axis B1 exponential shape at least from – 1.7 to 1 B1 not below x-axis (b) –1.41 (4…), 1.41 (4…) B1 B1 (c) –1.53 ( –1.532 to –1.531) B1 (d) 4 25 .0 ≤ ≤y B1B1 Condone < and allow in words. Allow f(x) or x for y. M1 for 0.25 and 4 soi. (3.75 implies this M1) [9]

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 31 © University of Cambridge International Examinations 2011 13 (a) 10 3 , 10 7 , 10 4 , 11 4 B2 Throughout this question allow decimal or percentage equivalents (at least 3 sf) but ratios or words score 0. Penalise 2 sf once. isw any cancelling or converting. B1 for 2 or 3 correct (b) (i) 110 42 oe B2 0.382 or 0.3818…. If B0, M1 for 10 6 11 7 × (ii) 110 56 oe ft B3 ft 0.509(0) to 0.5091 ft their diagram M2 for 11 7 × their 10 4 + their 11 4 × their 10 7 M1 for one of these products [7]