Cambridge IGCSE Mathematics (with coursework) 0581 — 2011 Oct/Nov Paper 3 · Variant 1

This document consists of 16 printed pages. IB11 11_0581_31/FP © UCLES 2011 [Turn over *2318181036* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/31 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 0581/31/O/N/11 For Examiner's Use 1 (a) Write twenty five million in figures. Answer(a) [1] (b) Write the following in order of size, starting with the smallest. 3 2 65% 0.6 Answer(b) I I [1] (c) In a sale a coat costing $250 is reduced to $200. Find the percentage decrease in the cost. Answer(c) % [3] (d) Basketball Tennis 90° 150° Football NOT TO SCALE 120 students are asked to choose their favourite sport. The results are shown in the pie chart. Calculate the number of students who chose (i) basketball, Answer(d)(i) [1] (ii) football. Answer(d)(ii) [2]

3 © UCLES 2011 0581/31/O/N/11 [Turn over For Examiner's Use 2 The distance between Geneva and Gstaad is 150 km. (a) Write 150 in standard form. Answer(a) [1] (b) A car took 1 2 1 hours to travel from Geneva to Gstaad. Calculate the average speed of the car. Answer(b) km/h [1] (c) A bus left Gstaad at 10 15. It arrived in Geneva at 12 30. Calculate the time, in hours and minutes, that the bus took for the journey. Answer(c) h min [1] (d) Another bus left Geneva at 13 55. It travelled at an average speed of 60 km/h. Find the time it arrived in Gstaad. Answer(d) [2] (e) The distance of 150 km is correct to the nearest 10 km. Complete the statement for the distance, d km, from Geneva to Gstaad. Answer(e) Y d I [2]

4 © UCLES 2011 0581/31/O/N/11 For Examiner's Use 3 36 29 41 45 15 10 13 Use the numbers in the list above to answer all the following questions. (a) Write down (i) two even numbers, Answer(a)(i) , [1] (ii) two prime numbers, Answer(a)(ii) , [2] (iii) a square number, Answer(a)(iii) [1] (iv) two factors of 90 . Answer(a)(iv) , [2] (b) (i) Calculate the mean of the seven numbers. Answer(b)(i) [2] (ii) Find the median. Answer(b)(ii) [2] (iii) Find the range. Answer(b)(iii) [1]

5 © UCLES 2011 0581/31/O/N/11 [Turn over For Examiner's Use (c) A number from the list is chosen at random. Find the probability that the number is (i) even, Answer(c)(i) [1] (ii) a multiple of 5. Answer(c)(ii) [1]

6 © UCLES 2011 0581/31/O/N/11 For Examiner's Use 4 (a) Using the exchange rates $1 = 0.70 Euros and $1 = 90 Yen change (i) $100 to Euros, Answer(a)(i) Euros [1] (ii) 100 Yen to dollars. Answer(a)(ii) $ [2] (b) Tania went on holiday to Switzerland. The exchange rate was $1 = 1.04 Swiss francs (CHF). She changed $1500 to Swiss francs and paid 1% commission. (i) How much commission, in dollars, did she pay? Answer(b)(i) $ [1] (ii) Show that she received CHF 1544.40. Answer (b)(ii) [2] (c) Tania spent CHF 950 on her holiday. She converted the remaining Swiss francs back into dollars. She paid CHF 10 to make the exchange. Calculate the amount, in dollars, Tania received. Answer(c) $ [3]

7 © UCLES 2011 0581/31/O/N/11 [Turn over For Examiner's Use 5 y x 6 5 4 3 2 1 –1 –2 –3 0 1 2 3 4 5 6 –1 –2 –3 –4 l (a) Find the gradient of the line l. Answer(a) [2] (b) (i) Complete the table below for x + 2y = 6 . x 0 2 y 0 [3] (ii) On the grid, draw the line x + 2y = 6 for −4 Y x Y 6 . [2] (c) The equation of the line l is 4x + 3y = 4. Use your diagram to solve the simultaneous equations 4x + 3y = 4 and x + 2y = 6 . Answer(c) x = y = [2]

8 © UCLES 2011 0581/31/O/N/11 For Examiner's Use 6 (a) A B The line AB is drawn above. Parts (i), (iii), and (v) must be completed using a ruler and compasses only. All construction arcs must be clearly shown. (i) Construct triangle ABC with AC = 7 cm and BC = 6 cm. [2] (ii) Measure angle BAC. Answer(a)(ii) Angle BAC = [1] (iii) Construct the bisector of angle ABC. [2] (iv) The bisector of angle ABC meets AC at T. Measure the length of AT. Answer(a)(iv) AT = cm [1] (v) Construct the perpendicular bisector of the line BC. [2] (vi) Shade the region that is • nearer to B than to C and • nearer to BC than to AB. [1]

9 © UCLES 2011 0581/31/O/N/11 [Turn over For Examiner's Use (b) A ship sails 40 km on a bearing of 040° from P to Q. (i) Using a scale of 1 centimetre to represent 5 kilometres, make a scale drawing of the path of the ship. Mark the point Q. North P Scale: 1 cm = 5 km [2] (ii) At Q the ship changes direction and sails 30 km on a bearing of 160° to the point R. Draw the path of the ship. [2] (iii) Find how far, in kilometres, the ship is from the starting position P. Answer(b)(iii) km [1] (iv) Measure the bearing of P from R. Answer(b)(iv) [1]

10 © UCLES 2011 0581/31/O/N/11 For Examiner's Use 7 (a) Solve the equation 2(x + 4) = 3(x + 2) + 8 . Answer(a) x = [3] (b) Make z the subject of za + b = 3 . Answer(b) z = [2] (c) Find x when 2x3 = 54 . Answer(c) x = [2]

11 © UCLES 2011 0581/31/O/N/11 [Turn over For Examiner's Use (d) A rectangular field has a length of x metres. The width of the field is (2x – 5) metres. (i) Show that the perimeter of the field is (6x – 10) metres. Answer (d)(i) [2] (ii) The perimeter of the field is 50 metres. Find the length of the field. Answer(d)(ii) length = m [2]

12 © UCLES 2011 0581/31/O/N/11 For Examiner's Use 8 A B y x 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 The diagram shows two shapes A and B. (a) Describe fully the single transformation which maps A onto B. Answer(a) [2] (b) On the grid, draw the line x = 2. [1] (c) On the grid, draw the image of shape A after the following transformations. (i) Reflection in the line x = 2. Label the image C. [1] (ii) Enlargement, scale factor 2, centre (0, 0). Label the image D. [2]

13 © UCLES 2011 0581/31/O/N/11 [Turn over For Examiner's Use 9 (a) Factorise completely 3x2 + 12x. Answer(a) [2] (b) Find the value of a3 + 3b2 when a = 2 and b = −2 . Answer(b) [2] (c) Simplify 3x4 × 2x3. Answer(c) [2]

14 © UCLES 2011 0581/31/O/N/11 For Examiner's Use 10 2 m 5 m 10 m x m NOT TO SCALE The diagram shows a ramp in the form of a triangular prism. The cross-section is a right-angled triangle of length 5 m and height 2 m. (a) Find the value of x. Give your answer correct to 1 decimal place. Answer(a) x = [3] (b) Find the area of the cross-section. Answer(b) m2 [2] (c) The ramp is 10 m long. Calculate the volume of the ramp. Answer(c) m3 [1]

15 © UCLES 2011 0581/31/O/N/11 [Turn over For Examiner's Use (d) Calculate the total surface area of all five faces of the ramp. Answer(d) m2 [3] (e) Each face of the ramp is painted. Paint costs $2.25 per square metre. Calculate the total cost of the paint. Answer(e) $ [1] 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. © UCLES 2011 0581/31/O/N/11 For Examiner's Use 11 Diagram 1 Diagram 2 Diagram 3 The diagrams show a sequence of shapes. (a) On the grid, draw Diagram 4. [1] (b) Complete the table showing the number of lines in each diagram. Diagram (n) Number of lines 1 6 2 11 3 4 5 [3] (c) Work out the number of lines in Diagram 8. Answer(c) [1] (d) Write down an expression, in terms of n, for the number of lines in Diagram n. Answer(d) [2] (e) Work out the number of lines in Diagram 100. Answer(e) [1] (f) The number of lines in Diagram p is 66. Find the value of p. Answer(f) p = [2]

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 0581/31 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 31 © 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) 25 000 000 cao 1 (b) 0.6 < 65% < 3 2 1 (c) 20% 3 B1 for 50 seen M1 for 250 50 their × 100 or B1 for 0.8 or 80 seen M1 for 1 – their 0.8 or 100 – their 80 (d) (i) 30 1 (ii) 40 2 M1 for 360 – (90 + 150) implied by 120 seen 2 (a) 1.5(0) × 102 cao 1 (b) 100 cao 1 (c) 2 hours 15 minutes cao 1 (d) 16(:) 25 (pm) or (0)425 pm 2 M1 for 2.5 (oe), 2hrs 30 min (e) 145 ≤ d < 155 2 B1 for each value in correct place 3 (a) (i) 36, 10 1 (ii) 29, 41, 13 any two 2 B1 for each (iii) 36 1 (iv) 45, 15, 10 any two 2 B1 for each (b) (i) 27 2 B1 for 36 + 29 + … + 13 seen implied by 189 (ii) 29 2 M1 for attempting to order the numbers (iii) 35 cao 1 (c) (i) 7 2 oe 1 (ii) 7 3 oe 1ft Their denominator from (c)(i)

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 31 © University of Cambridge International Examinations 2011 4 (a) (i) 70 cao 1 (ii) 1.11(11…) 2 B1 for 100 ÷ 90, 10 ÷ 9, 1 9 1 (b) (i) 15 cao 1 (ii) (1500 – 15) × 1.04 2 B1 for × 1.04, 1560, 15.60 (c) 561.92 3 M1 for 1544.40 – 950 – 10 (584.40) oe M1 indep for ÷ 1.04 5 (a) 3 4 − oe, –1.2 to –1.4 2 B1 for attempt at run rise (b) (i) 3, 2, 6 3 B1 for each value (ii) Correct continuous line 2ft Minimum length (0,3) to (6,0) B1 for plotting their 3 points (c) x = −2, y = 4 2ft B1 for their x, B1 for their y from their intersections 6 (a) (i) Correct construction 2 B1 for two lines or B1 for accurate arcs seen or B1 for one correct line with two arcs SC1 for AC = 6 and BC = 7 with arcs (ii) 47° (45 – 49) 1ft Strict ft their (a)(i) (iii) Correct construction 2ft Their (a)(i) B1 for accurate arcs no line or B1 for accurate line drawn no arcs or B1 for accurate line with arcs bisecting another angle (iv) 4 (3.8 – 4.2) 1ft Strict ft their (iii) with intersection on opposite side of triangle (v) Correct construction 2ft B1 for accurate arcs no line or B1 for accurate line drawn no arcs or B1 for accurate line with arcs, bisecting AB or AC (vi) Correct region shaded 1ft ft is for boundaries of correct perpendicular bisector of their BC and correct angle bisector of their ABC, with or without arcs (b) (i) Correct scale drawing of PQ 2 B1 for accurate angle 40o, B1 for PQ 8cm (ii) Correct scale drawing of their QR 2 B1 for accurate angle 160o, B1 for QR 6cm (iii) 35 to 37 1ft Measure × 5 ± 1km (iv) 264 to 268 1ft

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2011 0581 31 © University of Cambridge International Examinations 2011 7 (a) −6 www 3 M2 for 8 = x + 6 + 8 or better or –x + 8 = 6 + 8 or better M1 for 2x + 8 or 3x + 6 or 3x + 14 (b) a b − 3 or a b a − 3 2 B1 for 3 – b seen or a a b z 3 = + (c) 3 2 B1 for 2 54 or better SC1 for embedded answer ie 2 × 33 = 54 or 2 × 3 × 3 × 3 = 54 (d) (i) x + x + 2x − 5 + 2x − 5 = 6x – 10 2 M1 accept 2x + 2(2x – 5) or 2(x + 2x – 5) E1 dep (ii) 10 2 M1 for 6x – 10 = 50 8 (a) Translation       −6 0 2 B1 for translation B1 for column vector (b) Correct line drawn 1 Continuous full line. Accept freehand. (c) (i) Correct reflection 1ft Their (b) (ii) Correct enlargement 2 B1 for any other enlargement scale factor 2 9 (a) 3x(x + 4) 2 B1 for 3(x2 + 4x) or B1 for x(3x + 12) or B1 for 3x(x + 4) seen (if not final answer) (b) 20 2 B1 for 8 or 12 seen (c) 6x7 2 B1 for kx7 or for 6xk , k ≠ 0 10 (a) 5.4 cao 3 M1 for 22 + 52 (= x2) implied by 29 A1 5.38(51..) or 29 or 5.39 B1 indep for rounding their answer to 1 decimal place (b) 5 2 M1 for 0.5 × 5 × 2 oe (c) 50 1ft 10 × their (b) (d) 134 3ft M2 for 2 × their (b) + 10 × their (a) + 2 × 10 + 5 × 10 or better M1 for any 3 faces correct (e) 301.5(0) 1ft Their (d) × 2.25 11 (a) Correct shape drawn 1 (b) 16, 21, 26 3 B1 for each SC1 “their 16” + 5 SC1 “their 21” + 5 (c) 41 1 (d) 5n + 1 2 B1 for 5n, B1 for +1 (e) 501 1ft Their (d) if linear (f) 13 2ft Their (d) if linear B1 for their (d) = 66