Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 Oct/Nov Paper 3 · Variant 2

This document consists of 16 printed pages. IB11 11_0581_32/RP © UCLES 2011 [Turn over *4745867181* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/32 Paper 3 (Core) October/November 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 0581/32/O/N/11 For Examiner's Use 1 Mr and Mrs Sayed and their 3 children go on holiday. They travel to the airport by train. (a) The train departs at 16 20. (i) They leave home 45 minutes before the train departs. Find the time at which they leave home. Answer(a)(i) [1] (ii) Write 16 20 using the 12-hour clock. Answer(a)(ii) [1] (b) The train fare is $24 for an adult. The train fare for a child is 3 2 of an adult fare. Find (i) the fare for a child, Answer(b)(i) $ [1] (ii) the total fare for Mr and Mrs Sayed and their 3 children. Answer(b)(ii) $ [2]

3 © UCLES 2011 0581/32/O/N/11 [Turn over For Examiner's Use 2 Aminata buys a business costing $23 000. (a) She pays part of this cost with $12 000 of her own money. Calculate what percentage of the $23 000 this is. Answer(a) % [1] (b) Aminata’s brother gives her 32% of the remaining $11 000. Show that $7 480 is still needed to buy the business. Answer(b) [2] (c) Aminata borrows the $7 480 at a rate of 3.5 % per year compound interest. Calculate how much money she owes at the end of 3 years. Answer(c) $ [3] (d) In the first year Aminata spent $11 000 on salaries, equipment and expenses. 5 2 of this money was spent on salaries, 0.45 of this money was spent on equipment and the remainder was for expenses. Calculate how much of the $11 000 was spent on (i) salaries, Answer(d)(i) $ [1] (ii) equipment, Answer(d)(ii) $ [1] (iii) expenses. Answer(d)(iii) $ [1] (e) The three items in part (d) are in the ratio salaries : equipment : expenses = 0.4 : 0.45 : 0.15 . Write this ratio in its simplest form. Answer(e) : : [2]

4 © UCLES 2011 0581/32/O/N/11 For Examiner's Use 3 (a) r =       −2 3 +       − − 2 5 (i) Write down r as a single vector. Answer(a)(i) r =           [1] (ii) The point G(3, 2) is translated by the vector r to the point H. Find the co-ordinates of H. Answer(a)(ii) ( , ) [1] (iii) Write down the vector of the translation that maps H onto G. Answer(a)(iii)           [1]

5 © UCLES 2011 0581/32/O/N/11 [Turn over For Examiner's Use (b) P Q y x 10 9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 0 1 2 3 4 5 6 7 8 9 10 –1 –2 –3 –4 The diagram shows two triangles P and Q. (i) Describe fully the single transformation which maps P onto Q. Answer(b)(i) [3] (ii) On the grid, draw the reflection of P in the line x = 0. Label this image R. [2] (iii) On the grid, rotate P through 180° about (0, 0). Label this image S. [2] (iv) Describe fully the single transformation which maps triangle S onto triangle R. Answer(b)(iv) [2]

6 © UCLES 2011 0581/32/O/N/11 For Examiner's Use 4 (a) Expand and simplify 3(2x + y) + 5(x – y). Answer(a) [2] (b) Expand x2(3x – 2y). Answer(b) [2] (c) Factorise completely 4y2 – 10xy. Answer(c) [2] (d) y = 3 4 2 x (i) Find the value of y when x = O3 . Answer(d)(i) y = [2] (ii) Make x the subject of the formula. Answer(d)(ii) x = [3]

7 © UCLES 2011 0581/32/O/N/11 [Turn over For Examiner's Use 5 (a) An aeroplane takes off 140 metres before reaching the end of the runway. It climbs at an angle of 22° to the horizontal ground. 22° h 140 m NOT TO SCALE Calculate the height of the aeroplane, h, when it is vertically above the end of the runway. Answer(a) h = m [2] (b) After 3 hours 30 minutes the aeroplane has travelled 1850 km. Calculate the average speed of the aeroplane. Answer(b) km/h [2] (c) 15 km NOT TO SCALE A B C The aeroplane descends from A, at a height of 12 000 metres, to C, at a height of 8 300 metres. (i) Work out the vertical distance, BC, that the aeroplane descends. Answer(c)(i) m [1] (ii) The distance AC is 15 kilometres. Calculate angle BAC. Answer(c)(ii) Angle BAC = [2]

8 © UCLES 2011 0581/32/O/N/11 For Examiner's Use 6 F E A D B C 24 cm 30 cm 16 cm NOT TO SCALE The diagram shows a wedge in the shape of a triangular prism. AB = 30 cm, AF = 16 cm and BC = 24 cm. Angle BAF = 90°. (a) Calculate (i) the area of triangle ABF, Answer(a)(i) cm2 [2] (ii) the volume of the wedge. Answer(a)(ii) cm3 [1] (b) (i) Calculate BF. Answer(b)(i) cm [2] (ii) 1.6 cm NOT TO SCALE A coin with diameter 1.6 cm is rolled down the sloping surface of the wedge. It travels in a straight line parallel to BF, starting on FE and ending on BC. Calculate the number of complete turns it makes. Answer(b)(ii) [3]

9 © UCLES 2011 0581/32/O/N/11 [Turn over For Examiner's Use (c) On the grid, complete the net of the wedge. The base and one of the triangles have been drawn for you. Each square on the grid represents a square of side 4 centimetres. [3] (d) Calculate the surface area of the wedge. Answer(d) cm2 [3]

10 © UCLES 2011 0581/32/O/N/11 For Examiner's Use 7 (a) The table shows some values for y = x 18 . x O9 O6 O4 O3 O2 2 3 4 6 9 y O2 O4.5 O9 4.5 3 (i) Complete the table. [2] (ii) On the grid, draw the graph of y = x 18 for O9 Y x Y O2 and 2 Y x Y 9 . y x 9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9 0 –1 –2 –3 –4 –5 –6 –7 –8 –9 4 5 6 7 8 9 3 2 1 [4] (iii) Use your graph to solve the equation x 18 = O5 . Answer(a)(iii) x = [1]

11 © UCLES 2011 0581/32/O/N/11 [Turn over For Examiner's Use (b) (i) Complete the table of values for y = 2x + 3 . x O4 O3 2 3 y O5 7 [2] (ii) On the grid, draw the graph of y = 2x + 3 for O4 Y x Y 3 . [1] (iii) Find the co-ordinates of the points of intersection of the graphs of y = x 18 and y = 2x + 3 . Answer(b)(iii) ( , ) and ( , ) [2]

12 © UCLES 2011 0581/32/O/N/11 For Examiner's Use 8 The table shows the average temperature and rainfall each month at Wellington airport. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Temperature (°C) 18 18 17 14 12 10 9 10 11 13 15 16 Rainfall (mm) 67 48 76 87 99 113 111 106 82 81 74 74 (a) Complete the bar chart to show the temperature each month. 20 18 16 14 12 10 8 6 4 2 0 Jan Feb Mar Apr May Jun Jul Month Aug Sep Oct Nov Dec Temperature (°C) [2] (b) For the rainfall calculate (i) the mean, Answer(b)(i) mm [2] (ii) the median. Answer(b)(ii) mm [2]

13 © UCLES 2011 0581/32/O/N/11 [Turn over For Examiner's Use (c) In the scatter diagram the rainfall for January to April is plotted against temperature. 120 115 110 105 100 95 90 85 80 75 70 65 60 55 50 45 40 8 9 10 11 12 13 14 15 16 17 18 19 20 Temperature (°C) Rainfall (mm) (i) Complete the scatter diagram by plotting the values for the months May to December. [3] (ii) Draw the line of best fit on the scatter diagram. [1] (iii) What type of correlation does the scatter diagram show? Answer(c)(iii) [1]

14 © UCLES 2011 0581/32/O/N/11 For Examiner's Use 9 On the scale drawing opposite, point A is a port. B and C are two buoys in the sea and L is a lighthouse. The scale is 1 cm = 3 km. (a) A boat leaves port A and follows a straight line course that bisects angle BAC. Using a straight edge and compasses only, construct the bisector of angle BAC on the scale drawing. [2] (b) When the boat reaches a point that is equidistant from B and from C, it changes course. It then follows a course that is equidistant from B and from C. (i) Using a straight edge and compasses only, construct the locus of points that are equidistant from B and from C. Mark the point P where the boat changes course. [2] (ii) Measure the distance AP in centimetres. Answer(b)(ii) cm [1] (iii) Work out the actual distance AP. Answer(b)(iii) km [1] (iv) Measure the obtuse angle between the directions of the two courses. Answer(b)(iv) [1] (c) Boats must be more than 9 kilometres from the lighthouse, L. (i) Construct the locus of points that are 9 kilometres from L. [2] (ii) Mark the point R where the course of the boat meets this locus. Work out the actual straight line distance, AR, in kilometres. Answer(c)(ii) km [1]

15 © UCLES 2011 0581/32/O/N/11 [Turn over For Examiner's Use A B C L Scale: 1 cm = 3 km 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 0581/32/O/N/11 For Examiner's Use 10 (a) Write down the next term in each of the following sequences. (i) 2, 9, 16, 23, [1] (ii) 75, 67, 59, 51, [1] (iii) 2, 5, 9, 14, [1] (iv) 2, 1, 2 1 , 4 1 , [1] (v) 2, 4, 8, 16, [1] (b) For the sequence in part (a)(i) write down (i) the 10th term, Answer(b)(i) [1] (ii) the nth term. Answer(b)(ii) [2] (c) The nth term of the sequence in part (a)(iii) is 2 3 2 n n + . Calculate the 50th term of this sequence. Answer(c) [2] (d) The nth term of the sequence in part (a)(v) is 2n. Calculate the 12th term of this sequence. Answer(d) [1]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2011 question paper for the guidance of teachers 0581 MATHEMATICS 0581/32 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 October/November 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 – October/November 2011 0581 32 © 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 Marks 1 (a) (i) 15 35 1 Accept 3.35 pm Condone 1535 pm (ii) (0)4 20 pm cao 1 (b) (i) 16(.00) 1 (ii) 96(.00) 2 M1 for 2 × 24 + 3 × their (b)(i) seen or implied 2 (a) 52.2(%) or 52.17… 1 (b) 11000 − (32 ÷ 100 × 11000) or (68 ÷ 100 × 11000) M1 (=) 7480 E1 Must see this for the second mark. (c) 8293 or 8290 or 8293.2 or 8293.21 as final answer 3 Either M1 for 7480 × 1.0352 oe or M1 for 7480 × 1.035 = 7741.8 and their 7741.8 × 1.035 (M1 implied by 8012.76...) Then M1 dep for completion of method for the third year If zero SC1 for answer 813.(2…) (d) (i) 4 400 1 (ii) 4 950 1 (iii) 1 650 1ft 11 000 − their (d)(i) − their (d)(ii) (e) 8 : 9 : 3 cao 2 B1 for 40 : 45 : 15 oe seen or correct non-integer ratio

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 32 © University of Cambridge International Examinations 2011 3 (a) (i) (r =)       − − 4 2 1 (ii) (1, −2) 1ft (3 + their −2, 2 + their −4) (iii)       4 2 1ft Inverse of their (a)(i) (b) (i) Enlargement 1 All independent (Scale Factor) 3 1 (Centre) (0, 0) 1 (ii) Reflection in x = 0 drawn 2 SC1 Reflection in y = 0 (iii) Rotation 180° about (0, 0) drawn 2 SC1 180° rotation about any other point (iv) Reflection x axis or y = 0 1ft 1ft Strict follow through Independent marks 4 (a) 11x − 2y final answer 2 B1 for 6x + 3y or 5x − 5y or 11x or −2y in working (b) 3x3 − 2x2 y final answer 2 B1 for 3x3 ± jx2y or kx3 − 2x2y (c) 2y(2y − 5x) final answer 2 B1 for y(4y − 10x) or 2(2y2 − 5xy) or SC1 for 2y(2y + 5x) or SC1 for 2y(2y − 5x) in working but then spoilt (d) (i) 12 2 M1 for 3 ) 3 ( 4 2 − × or better in working. (ii) (x) = 4 3y final answer oe 3 Maximum of M2 from M1 for × by 3 M1 for ÷ by 4 M1 for square root 5 (a) 56.6 or 56.56… 2 M1 for tan 22 =140 h or better or M1 for tan(90–22) = h 140 or better (b) 529 (km/h) or 528.6 or 528.57… 2 M1 for 5.3 ) 1850 ( or better. (c) (i) 3700(m) 1 (ii) 14.3 or 14.2(8…) 2ft M1 for sin (BAC) = 15000 their (c)(i)

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 32 © University of Cambridge International Examinations 2011 6 (a) (i) 240 2 M1 for 0.5 × 30 × 16 (ii) 5760 1ft ft is (a)(i) × 24 (b) (i) 34 2 M1 for (FB 2) = 16 2 + 30 2 (ii) 6 3 M1 for (circumference) = 1.6 × π M1 dep their (b)(i) ÷ their 1.6π (6.76 implies M1, M1) If 0 scored either SC1 for their (b)(i) ÷ 3.2 × π and then SC1 for truncating correctly If M1 or still 0 scored then SC1 for truncating correctly any number with at least 1 decimal place (c) 6 by 4 rectangle above 1 6 by their 8.5 rectangle below 1ft ft (b)(i) ÷ 4 Correct triangle on AB 1 (d) 2400 3cao M2 for 2 1 × 30 × 16 + 2 1 × 30 × 16 + 16 × 24 + 30 × 24 + their 34 × 24 (M1 for any 3 areas) If 0, SC2 for 150 or SC1 for 120 (3 rectangles) or SC1 for 30 (2 triangles) 7 (a) (i) −3, −6, 9, 6, 2 2 B1 for 4 correct (ii) Graph P3ft P2ft for 8 or 9 points correct P1ft for 6 or 7 points correct C1 Correct curve and not crossing axis (iii) −3.7 to −3.5 1ft ft their curve (b) (i) −3, 9 1, 1 (ii) Ruled continuous line y = 2x + 3 1 Line long enough to intersect both parts (iii) (2.2 to 2.5, 7.5 to 7.8) 1ft ft their line intersection with the curves (−4.0 to −3.7, −4.8 to −4.5) 1ft 8 (a) heights 11, 13, 15, 16 2 B1 for 3 correct (b) (i) 84.8(3…) 2 M1 addition of 12 rainfall values (ii) 81.5 2 Either M1 for evidence of ordering values or substantial part of list (at least first 7 or last 7) or M1 for answers of 81 and 82 (c) (i) 8 values correctly plotted P3 P2 for 6 or 7 correct P1 for 4 or 5 correct (ii) Line of best fit 1 Must be continuous and straight (iii) Negative 1

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 32 © University of Cambridge International Examinations 2011 9 (a) Bisector of angle BAC with correct arcs 2 Either B1 correct without arcs or B1 for 2 pairs of accurate arcs seen (b) (i) Bisector of BC with 2 pairs of correct arcs 2 Either B1 correct without arcs or B1 for 2 pairs of accurate arcs seen (ii) 10.8 to 11.2 (cm) cao 1 (iii) 32.4 to 33.6 1ft Their (b)(ii) × 3 (iv) 155° to 165° cao 1 (c) (i) Circle centre L, radius 3cm 2 B1 circle centre L, incorrect radius or SC1 for part circle with correct radius (ii) 41km to 44km cao 1 10 (a) (i) 30 1 (ii) 43 1 (iii) 20 1 (iv) 8 1 or 0.125 1 (v) 32 1 (a) (i) 65 1 (ii) 7n − 5 or equivalent 2 B1 for 7n seen (c) 1325 2 B1 for 2 50 3 502 × + or better seen (d) 4096 1