Cambridge IGCSE Mathematics - International 0607 — 2011 May/June Paper 3 · Variant 2

This document consists of 17 printed pages and 3 blank pages. IB11 06_0607_32/3RP © UCLES 2011 [Turn over *5871924244* For Examiner's Use UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education CAMBRIDGE INTERNATIONAL MATHEMATICS 0607/32 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 0607/32/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 0607/32/M/J/11 [Turn over For Examiner's Use Answer all the questions. 1 (a) Write the ratio 12 : 20 in its simplest form. Answer(a) : [1] (b) x : 8 = 3 : 2. Find the value of x. Answer(b) x = [1] (c) Divide $30 in the ratio 3 : 7. Answer(c) $ : $ [2] (d) Write the fraction 2 6 3 xy xy in its lowest terms. Answer(d) [2] (e) Work out 7 19 of $570. Answer(e) $ [1] (f) Calculate 15% of 60 kg. Answer(f) kg [2] (g) Sam spends $6 at a shop. This is 3 25 of Sam’s pocket money. Calculate Sam’s pocket money. Answer(g) $ [2]

4 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 2 Ten students have the following shoe sizes. 33 36 36 33 35 37 33 32 38 38 (a) Find (i) the mode, Answer(a)(i) [1] (ii) the median, Answer(a)(ii) [1] (iii) the range, Answer(a)(iii) [1] (iv) the upper quartile, Answer(a)(iv) [1] (v) the mean. Answer(a)(v) [1] (b) 3 2 1 0 Shoe size Frequency On the grid, draw an accurate bar graph to show the frequencies of the shoe sizes. [3]

5 © UCLES 2011 0607/32/M/J/11 [Turn over For Examiner's Use (c) 35 32 33 Complete the pie chart accurately to show the frequencies of the remaining shoe sizes. Label your sectors clearly. [3] (d) Find the probability that a student chosen at random has a shoe size (i) greater than 36, Answer(d)(i) [1] (ii) greater than 30. Answer(d)(ii) [1] (e) Find the percentage of those students with a shoe size greater than 32 who have a shoe size greater than 33. Answer(e) % [2]

6 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 3 U = { x | 1 Y x Y 10, x ∈ w } A = { even numbers } B = { factors of 36 } (a) U A B Write the ten members of U in the correct regions of the Venn diagram. [3] (b) Complete the following. (i) A ∩ B = { } [1] (ii) A ∪ B = { } [1] (iii) A' ∩ B = { } [1] (iv) n (B' ) = [1]

7 © UCLES 2011 0607/32/M/J/11 [Turn over For Examiner's Use 4 B A U W T 9 m 6.5 m 57° 8 m NOT TO SCALE The diagram shows a vertical flagpole standing on horizontal ground. Two straight wires AT and BU are attached to the flagpole as shown. (a) U is 6.5 m above the ground and BU = 9 m. Calculate the angle between BU and the ground. Answer(a) [2] (b) The point W is vertically below the point T. AW = 8 m and angle WAT = 57°. Calculate TW, the height of the flagpole. Answer(b) m [2]

8 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 5 B O A 6 cm 6 cm NOT TO SCALE The diagram shows a sector OAB of a circle, radius 6 cm. Angle AOB = 90°. (a) Calculate (i) the area of triangle OAB, Answer(a)(i) cm2 [2] (ii) the area of the sector OAB, Answer(a)(ii) cm2 [2] (iii) the area of the region shaded in the diagram. Answer(a)(iii) cm2 [1]

9 © UCLES 2011 0607/32/M/J/11 [Turn over For Examiner's Use (b) Calculate (i) the length of AB, Answer(b)(i) cm [2] (ii) the perimeter of the region shaded in the diagram. Answer(b)(ii) cm [3]

10 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 6 D A C E B Q 80° NOT TO SCALE In the diagram, DEQ is parallel to AC. DE = BE and angle ACB = 80°. (a) (i) Write down the size of angle CEQ. Answer(a)(i) Angle CEQ = [1] (ii) Give a reason for your answer. [1] (b) Find the size of (i) angle BEQ, Answer(b)(i) Angle BEQ = [1] (ii) angle DBE, Answer(b)(ii) Angle DBE = [1] (iii) angle BAC. Answer(b)(iii) Angle BAC = [1]

11 © UCLES 2011 0607/32/M/J/11 [Turn over For Examiner's Use 7 y x 4 3 2 1 –1 –2 –3 –4 –4 –3 –2 –1 1 0 2 3 4 A B C l (a) Write down the co-ordinates of the point A. Answer(a) ( , ) [1] (b) Write the vector AB → in component form. Answer(b)           [1] (c) The line l passes through B and C. (i) Find the gradient of the line l. Answer(c)(i) [2] (ii) Find the equation of the line l. Answer(c)(ii) [2]

12 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 8 y x 20 –20 –4 6 0 (a) On the axes, sketch the graph of 10 3 y x = − , between x = – 4 and x = 6. (The graph has two separate parts.) [3] (b) The graph has two asymptotes. (i) On the axes, sketch the asymptote which is parallel to the y-axis. [1] (ii) Write down the equation of this asymptote. Answer(b)(ii) [1] (c) (i) On the axes, sketch the graph of 2 2 x y = . [1] (ii) Solve the equation 2 10 3 2 x x = − . Answer(c)(ii) x = [1]

13 © UCLES 2011 0607/32/M/J/11 [Turn over For Examiner's Use 9 6 cm 16 cm NOT TO SCALE The diagram shows a glass jug. The jug is a cylinder of radius 6 cm and height 16 cm. (a) Calculate the volume of the jug (i) in cm3, Answer(a)(i) cm3 [2] (ii) in litres. Answer(a)(ii) litres [1] (b) There are 1500 cm3 of water in the jug. (i) Calculate the height of the water in the jug. Answer(b)(i) cm [2] (ii) How many 25 cl glasses can be filled from the 1500 cm3 of water in the jug? Answer(b)(ii) [2]

14 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 10 (a) –3 –2 –1 0 1 2 x Write down the inequality shown in the diagram. Answer(a) [2] (b) Solve the simultaneous equations. Show your method. 2x + y = 1 y = 2x – 5 Answer(b) x = y = [3] (c) The perimeter, P, of a semicircle of radius r, is given by the following formula. P = πr + 2r (i) Factorise πr + 2r. Answer(c)(i) [1] (ii) Rearrange the formula P = πr + 2r to give r in terms of π and P. Answer(c)(ii) r = [1]

15 © UCLES 2011 0607/32/M/J/11 [Turn over For Examiner's Use 11 On any day, in February, the probability that it will rain in Hokitika is 3 7 . (a) For how many of the 28 days in February, would you expect it to rain? Answer(a) [1] (b) (i) Complete the tree diagram for two consecutive days by putting the probabilities in the spaces. First day Second day Rain No rain Rain No rain No rain Rain 3 7 3 7 [2] (ii) Calculate the probability that it will rain on two consecutive days. Answer(b)(ii) [2] (iii) Calculate the probability that it will rain on exactly one of the two days. Answer(b)(iii) [3] (iv) Complete the statement. The probability that [1] is 16 49 .

16 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 12 100 students estimate the length, l, of a piece of string. The results are shown in the table. Length (l cm) 0 Y l I 30 30 Y l I 40 40 Y l I 50 50 Y l I 60 60 Y l I 70 70 Y l I 100 Frequency 3 12 30 35 18 2 (a) Using the mid-values of the class intervals, calculate an estimate of the mean. Answer(a) cm [2] (b) (i) Complete the cumulative frequency table. Length (l cm) l I 30 l I 40 l I 50 l I 60 l I 70 l I 100 Cumulative frequency 3 15 98 100 [2] (ii) On the grid opposite, complete the cumulative frequency curve.

17 © UCLES 2011 0607/32/M/J/11 For Examiner's Use 100 90 80 70 60 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 0 Cumulative frequency Length (l cm) [2] (iii) Use your cumulative frequency curve to find the inter-quartile range. Answer(b)(iii) cm [2]

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20 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 0607/32/M/J/11 BLANK PAGE

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 0607/32 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 32 © University of Cambridge International Examinations 2011 1 (a) 3 : 5 B1 (b) 12 B1 (c) 9, 21 B1 B1 If B0, M1 for 30 ÷ 10 seen (not implied by 3) Condone 21, 9 (d) y 2 B2 B1 for 2 and x’s cancelled B1 independent for denominator y (e) 210 B1 (f) 9 B2 If B0, M1 for 0.15 × 60 oe (g) 50 B2 If B0, M1 for 6 ÷ 3 (implied by 2) seen [11] 2 (a) (i) 33 B1 (ii) 35.5 B1 (iii) 6 B1 (iv) 37 B1 (v) 35.1 B1 (b) Correct values on shoe axis Six correct heights (1, 3, …, 1, 2, 1, 2) B1 B2 i.e. labels not attached to grid lines. Condone absence of 34. B1 for five correct heights (c) Angles of 72°, 36° and 72° ( ± 2°) 3 correct labels of shoe sizes ft B2 ft B1 ft B1 for 1 correct ft their (b) ft their (b) (d) (i) 0.3 oe ft B1 ft ft their (b) or correct (ii) 1 oe B1 Allow 10 10 etc (e) 3 2 66 or 6. 66 & or 66.7 (or 66.66 to 66.67) ft B2 ft Accept 67. If B0, M1 for 6 ÷ 9 soi ft their (b) [15] 3 (a) U A B 8 10 2 4 6 1 3 9 5 7 B3 B2 for 9 correct, B1 for 8 correct (b) (i) 2, 4, 6 ft B1 ft (ii) 1, 2, 3, 4, 6, 8, 9, 10 ft B1 ft (iii) 1, 3, 9 ft B1 ft (iv) 4 ft B1 ft [7] 4 (a) 46.2 (46.23 to 46.24) B2 If B0, M1 for sin = 9 5.6 oe (b) 12.3 (12.31 to 12.32) B2 If B0, M1 for tan57 = 8 TW oe or better [4]

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 32 © University of Cambridge International Examinations 2011 5 (a) (i) 18 B2 If B0, M1 for 0.5 × 6 × 6 soi (ii) 28.3 (28.26 to 28.28) B2 If B0, M1 for 0.25 × π × r2 soi (iii) 10.3 (10.26 to 10.28) ft B1 ft ft their (ii) – their (i) (b) (i) 8.49 (8.485….) B2 If B0, M1 for 62 + 62 (ii) 17.9 (17.90 to 17.92) ft B3 ft ft 9.42 to 9.43 + their (i) If B0, M1 for 0.25 × π × 2r then M1 (dependent) for adding (i) [10] 6 (a) (i) 80 B1 (ii) Alternate or Z or diagram showing Z B1 (b) (i) 100 B1 (ii) 50 B1 (iii) 50 B1 [5] 7 (a) (3, –4) B1 (b)      − 5 3 B1 (c) (i) 3 2 B2 If B0, M1 for evidence of run rise (ii) 1 3 2 + = x y oe ft B2 ft Must be full equation ft their (c) If y = mx + c then B1 for x 3 2 and B1 (indep) for + 1 If c by ax = + oe, B2 for a, b, c B1 for 2 of them correct SC1 for 1 3 2 + x [6] 8 (a) Reasonable rectangular hyperbola shape Not touching x-axis x = 3 approximately looking an asymptote C1 B1 B1 Condone slight curving inwards from asymptotes Independent Independent and fairly generous (b) (i) Vertical asymptote drawn for their curve B1 Must look an asymptote but can be freehand (ii) x = 3 cao B1 (c) (i) U-shaped parabola, vertex at origin B1 (ii) 4.16 (or 4.157…) B1 [7] If graph is 3 10 − x ft as follows (a) C1, B0, B0 (b)(i) y-axis with some extra indication it is an asymptote B1 (ii) x = 0 B1 (c) (i) B1 (ii) 2 B1

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2011 0607 32 © University of Cambridge International Examinations 2011 9 (a) (i) 1808 to 1810 B2 If B0, M1 for π × 62 × 16 (ii) 1.808 to 1.81 ft B1 ft (b) (i) 13.3 (13.26 – 13.27) B2 If B0, M1 for π × 62 × h = 1500 o.e. (ii) 6 B2 If B0, SC1 for figs 6 [7] 10 (a) –2 ≤ x < 1 or x ≥ –2 and x < 1 B1 B1 SC1 for – 2 < x ≤ 1 (b) x = 1.5, y = –2 M1 A2 M1 for eliminating one variable to equation l kx = or l ky = 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) (c) (i) r(π + 2) B1 (ii) 2 + π P cao B1 [7] Throughout question 11, do not allow ratios or words. If decimals or percentages used, usual accuracy applies except penalise two sf by – 1 only once 11 (a) 12 B1 (b) (i) 7 4 , 7 3 , 7 4 , 7 4 against relevant branches B2 B1 for 2 or 3 correct (ii) 49 9 oe B2 (0.184 or 0.1836 to 0.1837) If B0, M1 for 7 3 7 3 × (iii) 49 24 oe B3 (0.49(0) or 0.4897 to 0.4898) If B0, M2 for 7 3 7 4 7 4 7 3 × + × o.e M1 for one of the products (0.24489…) (iv) It does not rain (on either day) oe B1 [9] 12 (a) 50.8 B2 If B0, M1 for at least 3 correct mid-values seen, not all from middle four (b) (i) 45, 80 B1 B1 (ii) (50, 45) and (60, 80) ft plotted Curve completed through 2 plotted points ft P1 ft C1 ft ft their table Only ft if correct shape maintained (iii) 14 to 16 ft B2 ft B1 for one correct quartile seen (42 to 44 or 57 to 59) ft their curve but only if curve increasing [8]