Cambridge IGCSE Mathematics (with coursework) 0581 — 2010 Oct/Nov Paper 3 · Variant 1

This document consists of 15 printed pages and 1 blank page. IB10 11_0581_31/FP © UCLES 2010 [Turn over *0449042992* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/31 Paper 3 (Core) October/November 2010 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 2010 0581/31/O/N/10 For Examiner's Use 1 (a) Write down (i) a multiple of 7 between 80 and 90, Answer(a)(i) [1] (ii) a prime number between 30 and 40, Answer(a)(ii) [1] (iii) a square number between 120 and 130, Answer(a)(iii) [1] (iv) a cube number between 100 and 200. Answer(a)(iv) [1] (b) Write the following numbers in order, starting with the smallest. 0.31 5 9 55% Answer(b) I I [2]

3 © UCLES 2010 0581/31/O/N/10 [Turn over For Examiner's Use 2 O S R P T 36° NOT TO SCALE The points P, R and S lie on a circle, centre O. ROT is a straight line and TS is a tangent to the circle at S. Angle STO = 36°. (a) Write down the size of angle TSO, giving a reason for your answer. Answer(a) Angle TSO = because [2] (b) (i) Calculate the size of angle TOS. Answer(b)(i) Angle TOS = [1] (ii) Show that angle OPR = 63°. Answer(b)(ii) [2] (c) (i) Write down the size of angle PRS. Answer(c)(i) Angle PRS = [1] (ii) Calculate the size of angle PSR. Answer(c)(ii) Angle PSR = [1]

4 © UCLES 2010 0581/31/O/N/10 For Examiner's Use 3 Month Total rainfall (mm) Average daily sunshine (hours) January 79 6 February 84 7 March 62 4.5 April 46 1.5 May 53 3.5 June 54 1.5 The table shows some data about rainfall and sunshine. (a) For the rainfall, calculate (i) the mean, Answer(a)(i) mm [2] (ii) the range. Answer(a)(ii) mm [1] (b) For the sunshine, find (i) the mode, Answer(b)(i) h [1] (ii) the median. Answer(b)(ii) h [2] (c) Dinesh draws a pie chart to display the rainfall data. Calculate the sector angle for February. Answer(c) [2]

5 © UCLES 2010 0581/31/O/N/10 [Turn over For Examiner's Use (d) Amalia draws a pictogram to display the sunshine data for January and February. January February March (i) Complete the key for the pictogram. [1] (ii) Complete the pictogram for March. [1] (e) Priya draws a scatter diagram to find the correlation between rainfall and sunshine for January to June. (i) Complete the scatter diagram below. January and February are plotted for you. 90 80 70 60 50 40 0 1 2 3 Average daily sunshine (hours) 4 5 6 7 Total rainfall (mm) [2] (ii) What type of correlation does the scatter diagram show? Answer(e)(ii) [1] represents

6 © UCLES 2010 0581/31/O/N/10 For Examiner's Use 4 D C B A M L X 7 cm 7 cm NOT TO SCALE In the diagram, ABCD is a square of side 7 cm. BLC and DMA are equilateral triangles. (a) Find the perimeter of the shape ABLCDM. Answer(a) cm [1] (b) (i) Write down the size of angle CBL. Answer(b)(i) Angle CBL = [1] (ii) Calculate the length of LX. Answer(b)(ii) LX = cm [2] (c) (i) Calculate the area of triangle BLC. Answer(c)(i) cm2 [2] (ii) Calculate the area of the shape ABLCDM. Answer(c)(ii) cm2 [2]

7 © UCLES 2010 0581/31/O/N/10 [Turn over For Examiner's Use 5 A shopkeeper buys cheese for $3.75 per kilogram and sells it for $5.10 per kilogram. (a) Calculate his percentage profit. Answer(a) % [3] (b) Mrs Garcia buys cheese from the shopkeeper. Calculate the number of grams of cheese she can buy for $2.04 . Answer(b) g [2] (c) The shopkeeper sells 7 kg of cheese and has 3 kg left. (i) He reduces his selling price of $5.10 per kilogram by 70%. Calculate the reduced price. Answer(c)(i) $ [2] (ii) He sells the 3kg of cheese at the reduced price. Calculate the total amount of money he receives by selling all the cheese. Answer(c)(ii) $ [2]

8 © UCLES 2010 0581/31/O/N/10 For Examiner's Use 6 (a) Complete the table of values for 4 y x = , x ≠ 0 . x −4 −3 −2 −1 − 0.5 0.5 1 2 3 4 y −1.3 −2 −8 8 4 2 [2] (b) On the grid below, draw the graph of 4 y x = , for – 4 Y x Y – 0.5 and 0.5 Y x Y 4. y x –1 –2 –3 –4 4 3 2 1 0 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 [4]

9 © UCLES 2010 0581/31/O/N/10 [Turn over For Examiner's Use (c) Complete the following statement. The point (−2.5, ) lies on the graph of 4 y x = . [1] (d) (i) On the grid, draw the line y = 5. [1] (ii) Use your graphs to solve the equation 4 5 x = . Answer(d)(ii) x = [1] (e) (i) On the grid, draw the straight line joining the points (− 0.5 , − 8 ) and ( 2 , 2 ). [2] (ii) Find the gradient of this line. Answer(e)(ii) [1] (iii) Write down the equation of this line in the form y = mx + c. Answer(e)(iii) y = [2]

10 © UCLES 2010 0581/31/O/N/10 For Examiner's Use 7 (a) Solve the equation. 4x + 3 = 2 + 6x Answer(a) x = [2] (b) Simplify. 7(3x – 4y) – 3(5x + 2y) Answer(b) [2] (c) Factorise completely. 6g2 – 3g3 Answer(c) [2]

11 © UCLES 2010 0581/31/O/N/10 [Turn over For Examiner's Use 8 P Q R y x 6 5 4 3 2 1 –1 –2 –3 –4 –5 0 –1 –2 –3 –4 –5 –6 –7 5 6 7 4 3 2 Shapes P, Q, and R are shown on the grid. (a) On the grid, draw the image of shape P after (i) a rotation through 180° about the origin, [2] (ii) a reflection in the line y = 3, [2] (iii) a translation by the vector 5 3 −       . [2] (b) Describe fully the single transformation which maps (i) shape P onto shape Q, Answer(b)(i) [2] (ii) shape P onto shape R . Answer(b)(ii) [3]

12 © UCLES 2010 0581/31/O/N/10 For Examiner's Use 9 L M R 210 km 325 km North NOT TO SCALE The diagram shows three islands, L, M and R. L is due west of M and R is due south of M. LM = 210 km and LR = 325 km. (a) Calculate the distance RM. Answer(a) RM = km [3] (b) (i) Use trigonometry to calculate angle LRM. Answer(b)(i) Angle LRM = [2] (ii) Find the bearing of L from R. Answer(b)(ii) [2]

13 © UCLES 2010 0581/31/O/N/10 [Turn over For Examiner's Use (c) (i) A ferry travels directly from M to L. It leaves M at 06 15 and arrives at L at 13 45. Calculate the average speed of the ferry in kilometres per hour. Answer(c)(i) km/h [2] (ii) The ferry then travels the 325 km from L to R at an average speed of 37 km/h. Calculate the time taken. Give your answer in hours and minutes, to the nearest minute. Answer(c)(ii) h min [3] (iii) The ferry leaves L at 14 00. Use your answer to part (c)(ii) to find the time it arrives at R. Answer(c)(iii) [1]

14 © UCLES 2010 0581/31/O/N/10 For Examiner's Use 10 Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Each of the diagrams above shows one small shaded square and a number of small unshaded squares. The diagrams form a sequence. (a) Complete Diagram 5. [1] (b) Complete the table. Diagram 1 2 3 4 5 50 n Total number of small squares 1 4 9 16 Number of small shaded squares 1 1 1 1 Number of small unshaded squares 0 3 8 15 [7] (c) Diagram p has 9999 small unshaded squares. Find p. Answer(c) p = [1]

15 © UCLES 2010 0581/31/O/N/10 For Examiner's Use 11 Roberto earns a total of $p per week. He works for t hours each week and is paid a fixed amount per hour. He also receives a bonus of $k every week. The formula for p is p = 8t + k. (a) Write down how much Roberto is paid per hour. Answer(a) $ [1] (b) (i) Find how much Roberto earns in a week when he works for 40 hours and his bonus is $35. Answer(b)(i) $ [2] (ii) Find how many hours Roberto works in a week when he earns $288 and his bonus is $24. Answer(b)(ii) h [3] (c) Make t the subject of the formula. Answer(c) t = [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 2010 0581/31/O/N/10 BLANK PAGE

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2010 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. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the October/November 2010 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 2010 0581 31 © UCLES 2010 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 art anything rounding to soi seen or implied Qu. Answers Mark Part Marks 1 (a) (i) 84 cao (ii) 31 or 37 cao (iii) 121 cao (iv) 125 cao 1 1 1 1 (b) 31 .0 9 5 % 55 < < oe for each term 2 M1 for all numbers written as decimals or for all numbers written as percentages 2 (a) 90° (Angle between) tangent and radius/ diameter (b) (i) 54° cao 1 1 dep 1 (ii) ) 54 180 ( 2 1 − × or 180 – 90 – 1/2 (180 – 126) or 54/2 followed by (180 – 90 – 27 oe) 2 M1 for using isosceles triangle POR or M1 for using isosceles triangle ROS then triangle PRS (c) (i) 90° cao (ii) 27° cao 1 1 3 (a) (i) 63 (ii) 38 cao (b) (i) 1.5 cao (ii) 4 (c) 80° (d) (i) 1 hour (ii) 4 and a half more suns drawn (e) (i) 4 correct plots (ii) Positive 2 1 1 2 2 1 1 2 1 M1 for their “378” ÷ 6 or SC1 for 333 seen B1 for attempt to order the numbers M1 for 84 ÷ their total × 360 Condone size, shape of suns B1 for 3 or 2 correct

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2010 0581 31 © UCLES 2010 4 (a) 42 (b) (i) 60° 1 1 (ii) 6.06(217…) 2 M1 ft for 30 cos 7 = x or 60 sin 7 = x or 60 tan 5.3 = x or 30 tan 5.3 = x or better (c) (i) 21.2 to 21.4 ft 2ft M1 for × × 7 2 1 their (b)(ii) oe (ii) 91.4 to 91.7 ft 2ft M1 ft 7 × 7 + 2 (their (c)(i)) or B1 for 49 5 (a) 36 (%) 3 M2 for 100 75 .3 75 .3 1.5 × − M1 for 75 .3 1.5 or 136% or 1.36 or 5.1 – 3.75 implied by 1.35 (b) 400 (c) (i) 1.53 (ii) 40.29 cao 2 2 2 M1 for 2.04 ÷ 5.1 implied by figs 4 M1 for (1 – 0.7) × 5.1 oe or 5.10 – (5.10 × 0.70) M1 for 7 × 5.1 + 3 × their (c)(i) or 35.7 + (3 × their (c)(i) evaluated) 6 (a) –1, –4, 1.3, 1 (b) 10 points plotted ½ small square accuracy smooth correct curves not across y-axis (c) –1.6 correct or ft (d) (i) y = 5 drawn (ii) (x =) 0.8 correct or ft (e) (i) Ruled line drawn from (–0.5, –8) to (2, 2) (ii) 4 cao (iii) y = 4x – 6 or y = their (e)(ii) x + their intercept or y = 4x + their intercept 2 P3ft C1 1ft 1 1ft 2 1 2ft B1 for –1 and 1 and B1 for –4 and 1.3 P2 for 8 or 9 points, P1 for 5 or 6 or 7 points ft from their graph ft from their graph B1 for ruled line drawn from either point not horizontal or vertical B1 ft y = 4x + k or y = their (e)(ii) x + k or y = jx – 6 or y = jx + their intercept 7 (a) 0.5 or 1/2 (b) 6x – 34y or 2(3x – 17y) (c) 3g²(2 – g) cao 2 2 2 M1 for collecting terms correctly B1 for 21x – 28y or B1 for –15x – 6y or B1 for 6x or B1 for –34y B1 for correct partial factorising

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – October/November 2010 0581 31 © UCLES 2010 8 (a) (i) Rotated 180° about origin (ii) Reflected in y = 3 2 2 B1 for correct shape and orientation in wrong position B1 for reflection in x = 3 or y = k (iii) Translated by    − 3 5 2 B1 for translation by    − k 5 or       3 k or     −5 3 (b) (i) Reflection x = –1 (ii) Enlargement only (sf) 3 (centre) (1, 3) 1 1 1 1 1 B1 for each Independent Independent 9 (a) 248 art 3 M2 for 2 2 210 325 − or better M1 for 325² = x² + 210² or better (b) (i) 40.3° art 2 M1 sin = 210 ÷ 325 or 325 their cos (a) = or (a) their 210 tan = (ii) 319.7(5)° or 320° (c) (i) 28 (ii) 8h 47min (iii) 22 47 or 10 47 pm 2ft 2 3 1ft M1 for 360 – their (b)(i) B1 for (time =) 7.5 or 7.30 or M1 for 210 ÷ their 7.5 M1 for 325 ÷ 37 A1 for 8.78(37…) B1 independent converting decimal time to minutes ft 1400 + their (c)(ii) 10 (a) 5 by 5 shape (b) First row 25 2500 n² Second row 1 1 1 Third row 24 2499 n² – 1 (c) 100 1 1, 1, 1 1 1, 1, 1 1 Independent All three Independent 11 (a) 8 (b) (i) 355 1 2 M1 for 8 × 40 + 35 seen or better (ii) 33 3 M2 for 8 ) 24 288 ( − or B1 for 264 seen (c) 8 k p t − = 2 B1 mark for a correct step