Cambridge IGCSE Mathematics (with coursework) 0581 — 2014 May/June Paper 3 · Variant 1

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 an HB pencil for any diagrams or graphs. Do not use staples, paper clips, glue or correction fl uid. 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 specifi ed in the question, and if the answer is not exact, give the answer to three signifi cant fi gures. 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. MATHEMATICS 0581/31 Paper 3 (Core) May/June 2014 2 hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Tracing paper (optional) Cambridge International Examinations Cambridge International General Certifi cate of Secondary Education This document consists of 15 printed pages and 1 blank page. [Turn over IB14 06_0581_31/FP © UCLES 2014 *4135805420* PAPA CAMBRIDGE

2 0581/31/M/J/14 © UCLES 2014 1 (a) The angles in a triangle are in the ratio 3 : 4 : 8 . (i) Show that the smallest angle of the triangle is 36°. Answer(a)(i) [2] (ii) Work out the other two angles of the triangle. Answer(a)(ii) … and … [2] (b) Another triangle ABC has angle BAC = 35° and angle ABC = 65°. (i) Using a protractor and straight edge complete an accurate drawing of the triangle ABC. The side AB has been drawn for you. A B [2] (ii) Measure the length, in centimetres, of the shortest side of your triangle. Answer(b)(ii) … cm [1] (c) A different triangle has base 7.0 cm and height 5.6 cm. Calculate the area of this triangle, giving the units of your answer. Answer(c) … … [3] __________________________________________________________________________________________

3 0581/31/M/J/14 © UCLES 2014 [Turn over 2 (a) From the integers 50 to 100, fi nd (i) a multiple of 43, Answer(a)(i) … [1] (ii) a factor of 165, Answer(a)(ii) … [1] (iii) an odd number that is also a square number, Answer(a)(iii) … [1] (iv) a number which is a square number and also a cube number. Answer(a)(iv) … [1] (b) (i) Find the square root of 5929. Answer(b)(i) … [1] (ii) Find the lowest common multiple of 24 and 30. Answer(b)(ii) … [2] (c) Elena goes on a journey to the North Pole. She leaves home at 7 am on 15 July and arrives at the North Pole at 10 pm on 27 July. How long, in days and hours, did her journey take? Answer(c) … days … hours [2] __________________________________________________________________________________________

4 0581/31/M/J/14 © UCLES 2014 3 S P T y x –2 2 0 4 6 8 1 3 5 7 –4 –6 –8 –1 –3 –5 –7 10 9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 The diagram shows two shapes, S and T, on a 1 cm2 grid. P is the point (–2, 0).

5 0581/31/M/J/14 © UCLES 2014 [Turn over (a) (i) Write down the mathematical name of shape S. Answer(a)(i) … [1] (ii) How many lines of symmetry does shape S have? Answer(a)(ii) … [1] (b) Describe the single transformation that maps shape S onto shape T. Answer(b) … … [2] (c) On the grid, (i) draw the refl ection of shape S in the y-axis, [2] (ii) draw the rotation of shape S about (0, 0) through 90° anti-clockwise. [2] (d) On the grid, draw the enlargement of shape S with scale factor 2 and centre P (–2, 0). Label the image E. [2] (e) (i) Work out the area of shape S. Answer(e)(i) … cm2 [2] (ii) How many shapes, identical to shape S, will fi ll shape E completely? Answer(e)(ii) … [1] (iii) Work out the area of shape E. Answer(e)(iii) … cm2 [1] __________________________________________________________________________________________

6 0581/31/M/J/14 © UCLES 2014 4 Denzil grows tomatoes. He selects a random sample of 25 tomatoes. The mass of each tomato, to the nearest 5 grams, is shown below. 55 65 50 75 65 80 70 70 55 60 70 60 65 50 75 65 70 75 80 70 55 65 70 80 55 (a) (i) Complete the frequency table. You may use the tally column to help you. Mass (grams) Tally Frequency 50 55 60 65 70 75 80 [2] (ii) Write down the mode. Answer(a)(ii) … g [1] (iii) Find the range. Answer(a)(iii) … g [1] (iv) Show that the mean mass is 66 g. Answer(a)(iv) [2]

7 0581/31/M/J/14 © UCLES 2014 [Turn over (b) Denzil picks 800 tomatoes. 4% of the 800 tomatoes are damaged. How many of these tomatoes are not damaged? Answer(b) … [2] (c) Denzil sells 750 of his tomatoes. (i) The mean mass of a tomato is 66 g. Calculate the mass of the 750 tomatoes in kilograms. Answer(c)(i) … kg [3] (ii) Denzil sells his tomatoes at $1.40 per kilogram. Calculate the total amount he receives from selling all the 750 tomatoes. Answer(c)(ii) $ … [1] (iii) The cost of growing these tomatoes was $33. Calculate his percentage profi t. Answer(c)(iii) … % [3] __________________________________________________________________________________________

8 0581/31/M/J/14 © UCLES 2014 5 Use a ruler and compasses only in parts (a), (c) and (d) of this question. Show all your construction arcs. A B C D E P 100 m 100 m 120 m 150 m Scale: 1 cm to 20 m Maria owns a farm. The scale drawing shows part of the boundary of the farm. The scale is 1 centimetre represents 20 metres.

9 0581/31/M/J/14 © UCLES 2014 [Turn over (a) The point F is such that AF = 140 m and EF = 160 m. Angle BAF and angle DEF are both obtuse angles. Complete the scale drawing of the farm boundary ABCDEF. [2] (b) Write down the name of the polygon ABCDEF. Answer(b) … [1] (c) (i) Construct the perpendicular bisector of the side CD. [2] (ii) Construct the bisector of angle ABC. [2] (iii) All the farm buildings are within a region that is ● nearer to C than to D and ● nearer to BC than to BA. Shade the region containing the farm buildings. [1] (d) A fence post, P, is shown on the boundary DE. (i) Construct the locus of points that are 50 m from P and also inside the farm boundary. [2] (ii) A region for keeping pigs is within 50 m of P and inside the farm boundary. Calculate the actual area for keeping pigs. Answer(d)(ii) … m2 [2] __________________________________________________________________________________________

10 0581/31/M/J/14 © UCLES 2014 6 (a) (i) Complete the table of values for y = x 8 , x ≠ 0 . x –8 –4 –2 –1 1 2 4 8 y –2 2 [3] (ii) On the grid, draw the graph of y = x 8 for –8 Ğ x Ğ –1 and 1 Ğ x Ğ 8 . y x 8 6 4 2 –2 –4 –6 –8 0 –2 –4 –6 –8 6 8 4 2 [4]

11 0581/31/M/J/14 © UCLES 2014 [Turn over (iii) Write down the order of rotational symmetry of your graph. Answer(a)(iii) … [1] (b) (i) Complete this table of values for y = 1.5x + 3 . x –6 –4 –2 0 2 y –6 3 [2] (ii) On the grid, draw the graph of y = 1.5x + 3 . [1] (c) Use your graphs to solve the equation x 8 = 1.5x + 3 . Answer(c) x = … or x = … [2] (d) Write down the gradient of the graph of y = 1.5x + 3 . Answer(d) … [1] __________________________________________________________________________________________

12 0581/31/M/J/14 © UCLES 2014 7 120 people are asked how they travel to work. The pie chart shows the results. Bus Car Cycle Walk (a) (i) Show that 45 people travel by car. Answer(a)(i) [2] (ii) A person is chosen at random from the 120 people. Find the probability that this person travels to work by bus or by car. Answer(a)(ii) … [2]

13 0581/31/M/J/14 © UCLES 2014 [Turn over (b) One year later, the same 120 people were again asked how they travel to work. Here is the information. Number of people Walk x Cycle 31 Bus 17 more than the number of people who walk Car 2 times the number of people who walk (i) Use this information to complete the following equation, in terms of x. … = 120 [3] (ii) Solve the equation to fi nd the number of people who walk to work. Answer(b)(ii) … [3] __________________________________________________________________________________________

14 0581/31/M/J/14 © UCLES 2014 8 (a) Write down an expression for the total mass of c cricket balls, each weighing 160 grams, and f footballs, each weighing 400 grams. Answer(a) … grams [2] (b) Expand and simplify. 3(2x – 5y) – 4(x – 2y) Answer(b) … [2] (c) Factorise completely. 5x2y – 20x Answer(c) … [2] (d) Solve the simultaneous equations. 3x + 4y = 7 4x – 3y = 26 Answer(d) x = … y = … [4] __________________________________________________________________________________________

15 0581/31/M/J/14 © UCLES 2014 [Turn over 9 (a) For these sequences, write down the next two terms and the rule for fi nding the next term. (i) 84, 75, 66, 57, . . . Answer(a)(i) … , … rule … [3] (ii) 2, 6, 18, 54, . . . Answer(a)(ii) … , … rule … [3] (b) For the sequence in part (a)(i), (i) write down an expression, in terms of n, for the n th term, Answer(b)(i) … [2] (ii) fi nd the 21st term. Answer(b)(ii) … [2] __________________________________________________________________________________________

16 0581/31/M/J/14 © UCLES 2014 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. 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. BLANK PAGE

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2014 series 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 should be read in conjunction with the question paper and the Principal Examiner Report for Teachers. Cambridge will not enter into discussions about these mark schemes. Cambridge is publishing the mark schemes for the May/June 2014 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components. PAPA CAMBRIDGE

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 31 © Cambridge International Examinations 2014 Abbreviations cao correct answer only dep dependent FT follow through after error isw ignore subsequent working oe or equivalent SC Special Case nfww not from wrong working soi seen or implied Question Answers Mark Part Marks 1 (a) (i) 8 4 3 3 + + or 8 4 3 180 + + M1 3 ÷ (15) × 180 or 15 3 180 × (= 36) M1 (ii) 48 [and] 96 1,1 One mark for each. If zero, SC1 for sum of both angles = 144. (b) (i) Angle BAC = 35 (±2º) B1 Angle ABC = 65 (±2º) and triangle completed B1 If zero SC1 for AC and BC reversed and triangle completed (ii) 4.45cm to 4.85cm 1 FT FT for their shortest side (c) 19.6 cao 2 M1 for 0.5 × 7 × 5.6 cm2 oe 1 2 (a) (i) 86 1 (ii) 55 1 (iii) 81 1 (iv) 64 1 (b) (i) 77 1 (ii) 120 2 B1 for any other multiple of 120 (c) 12 [days] 15 [hours] 1,1

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 31 © Cambridge International Examinations 2014 3 (a) (i) Parallelogram 1 (ii) 0 1 (b) Translation 1       −6 9 1 Independent Accept 9 right, 6 down (c) (i) (1, 4), (4, 4), (5, 2), (2, 2). 2 SC1 for reflection in x-axis (ii) (−4, −1), (−4, −4), (−2, −5), (−2, −2) 2 SC1 for rotation 90° clockwise or correct rotation any centre (d) (–6,8), (0,8), (–8,4), (–2,4) 2 SC1 for enlargement of S, scale factor 2, wrong position (e) (i) 6 2 M1 for 3 × 2 (ii) 4 1 (iii) 24 1FT FT their(e)(i) × their (e)(ii) Or FT area of their (d) if a parallelogram and not congruent to S. 4 (a) (i) 2, 4, 2, 5, 6, 3, 3 2 B1 for 5 or 6 correct Or 7 correct tallies if frequency column blank Or 7 correct frequencies in tally column (ii) 70 1FT (iii) 30 1 (iv) ∑(Frequency, f × mass, w) 1650 ÷ 25 M1 B1 7 items attempted and added or sum of 25 masses (b) 768 2 M1 for 0.96 × 800 oe

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 31 © Cambridge International Examinations 2014 (c) (i) 49.5 cao 3 M1 for figs 66 × 750 soi M1 for ÷ 1000 (ii) 69.3[0] 1 FT Their (c)(i) × 1.40 (iii) 110 3 M2 for 100 33 33 ) )( ( × − ii c their or M1 for their (c)(ii) − 33 Alternative method M2 for 100 33 ) )( ( × ii c their – 100 Or M1 for 33 ) )( ( ii c their 5 (a) Hexagon correct with arcs. AF = 7 cm (±2mm) EF = 8 cm (±2mm) 2 B1 for correct hexagon without arcs or one length correct with arcs. Or B1 for two correct arcs (b) Hexagon 1 (c) (i) Bisector of CD with 2 pairs of arcs 2 B1 for correct bisector with one pair or no arcs (ii) Bisector of angle ABC with 2 pairs of correct arcs. 2 B1 for bisector without 2 pairs of arcs (iii) Correct enclosed region shaded 1FT Their enclosed region provided at least 1 mark in each of parts (i) and (ii) (d) (i) Semi-circle radius 2.5cm (±2mm) from P and inside polygon 2 SC1 for arc centre P radius 2.5cm Or for arc inside polygon centre P touching boundaries twice or any circle centre P. (ii) 3930 or 3926 to 3928 2 M1 for (π × 50²) ÷ 2 oe 6 (a) (i) −1, −4, −8, 8, 4, 1. 3 1 for each symmetrical pair (ii) 8 points correctly plotted, within ½ square. 3FT B2FT for 6 or 7 correct Or B1 FT for 4 or 5 correct 2 smooth correct curves, not joined 1 (iii) 2 1

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 31 © Cambridge International Examinations 2014 (b) (i) −3 0 6 2 B1 for two correct (ii) Correct ruled line 1 (c) 1.4 to 1.6 and −3.6 to −3.4 1FT,1FT FT from their graph ±0.1 (d) 1.5 1 7 (a) (i) [Car angle =] 135 (± 2°) 135 ÷ 360 × 120 ( = 45 ) B1 M1 (ii) 3 2 or value from 0.658 to 0.675 2 B1 for angles of 238° to 242° or 79 to 81 people (b) (i) x + 31 + x + 17 + 2x [= 120] or better 3 B1 for x + 17 – seen together B1 for 2x (ii) 18 cao 3 M1 FT for their (4x + 48) [=120] or their 2x + x + x = 120 – 31 – 17 or better. M1FT for their (4x = 72) If zero SC2 for a correct numerical solution of their equation of equivalent difficulty. 8 (a) 160c + 400f final answer 2 B1 for 160c or 400f seen (b) 2x – 7y final answer www 2 B1 for 2 x or –7y or 6x – 15y or –4x + 8y www (c) 5x(xy – 4) final answer 2 B1 for ) 4 ( 5 2 x y x − or ) 20 5 ( − xy x

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 31 © Cambridge International Examinations 2014 (d) [x=] 5 [y=] – 2 4 M1 for correctly equating one set of coefficients M1 for correct method to eliminate one variable A1 for correct x or y If zero scored SC1 for 2 values satisfying one of the original equations Alternative method M1 for correct rearrangement of one equation x = (7 – 4y) ÷ 3 or y = (7 – 3x) ÷ 4 or x = (26 + 3y) ÷ 4 or y = (4x – 26) ÷ 3 M1 for correct substitution in other equation 4(7 – 4y) ÷ 3 – 3y = 26 4x – 3(7 – 3x) ÷ 4 = 26 3(26 + 3y) ÷ 4 + 4y = 7 3x + 4(4x – 26) ÷ 3 = 7 (7 – 4y) ÷ 3 = (26 + 3y) ÷ 4 (7 – 3x) ÷ 4 = (4x – 26) ÷ 3 A1 for correct x or y If zero scored SC1 for 2 values satisfying one of the original equations 9 (a) (i) 48, 39 1, 1FT FT 6th term = 5th term −9 Subtract 9 oe 1 (ii) 162, 486 1, 1FT FT 6th term = 5th term × 3 Multiply by 3 oe 1 (b) (i) 93 – 9n oe final answer 2 B1 for −9n + c or kn + 93, k ≠ 0 (ii) −96 cao 2 M1 for substitution of n = 21 into their linear expression