Cambridge IGCSE Mathematics (with coursework) 0581 — 2014 May/June Paper 4 · 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 130. MATHEMATICS 0581/41 Paper 4 (Extended) May/June 2014 2 hours 30 minutes 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 19 printed pages and 1 blank page. [Turn over IB14 06_0581_41/RP © UCLES 2014 *9772345347* PAPA CAMBRIDGE

2 0581/41/M/J/14 © UCLES 2014 1 A = 1 2 1 - 3 f p B = (–2 5) C = 2 5 - e o D = 2 0 0 2 f p (a) Work out, when possible, each of the following. If it is not possible, write ‘not possible’ in the answer space. (i) 2A Answer(a)(i) [1] (ii) B + C Answer(a)(ii) [1] (iii) AD Answer(a)(iii) [2] (iv) A–1, the inverse of A. Answer(a)(iv) [2] (b) Explain why it is not possible to work out CD. Answer(b) … [1] (c) Describe fully the single transformation represented by the matrix D. Answer(c) … … [3] __________________________________________________________________________________________

3 0581/41/M/J/14 © UCLES 2014 [Turn over 2 Ali leaves home at 10 00 to cycle to his grandmother’s house. He arrives at 13 00. The distance-time graph represents his journey. 10 00 11 00 12 00 13 00 14 00 15 00 16 00 17 00 40 30 20 10 0 Time Distance from home (km) (a) Calculate Ali’s speed between 10 00 and 11 30. Give your answer in kilometres per hour. Answer(a) … km/h [2] (b) Show that Ali’s average speed for the whole journey to his grandmother’s house is 12 km/h. Answer(b) [2] (c) Change 12 kilometres per hour into metres per minute. Answer(c) … m/min [2] (d) Ali stays for 45 minutes at his grandmother’s house and then returns home. He arrives home at 16 42. Complete the distance-time graph. [2] __________________________________________________________________________________________

4 0581/41/M/J/14 © UCLES 2014 3 (a) The running costs for a papermill are $75 246. This amount is divided in the ratio labour costs : materials = 5 : 1. Calculate the labour costs. Answer(a) $ … [2] (b) In 2012 the company made a profi t of $135 890. In 2013 the profi t was $150 675. Calculate the percentage increase in the profi t from 2012 to 2013. Answer(b) … % [3] (c) The profi t of $135 890 in 2012 was an increase of 7% on the profi t in 2011. Calculate the profi t in 2011. Answer(c) $ … [3] (d) 2 cm NOT TO SCALE 21 cm 30 cm Paper is sold in cylindrical rolls. There is a wooden cylinder of radius 2 cm and height 21 cm in the centre of each roll. The outer radius of a roll of paper is 30 cm. (i) Calculate the volume of paper in a roll. Answer(d)(i) … cm3 [3]

5 0581/41/M/J/14 © UCLES 2014 [Turn over (ii) The paper is cut into sheets which measure 21 cm by 29.7 cm. The thickness of each sheet is 0.125 mm. (a) Change 0.125 millimetres into centimetres. Answer(d)(ii)(a) … cm [1] (b) Work out how many whole sheets of paper can be cut from a roll. Answer(d)(ii)(b) … [4] __________________________________________________________________________________________

6 0581/41/M/J/14 © UCLES 2014 4 B T P x 4 9 11 6 – x In the Venn diagram, = {children in a nursery} B = {children who received a book for their birthday} T = {children who received a toy for their birthday} P = {children who received a puzzle for their birthday} x children received a book and a toy and a puzzle. 6 children received a toy and a puzzle. (a) 4 children received a book and a toy. 5 children received a book and a puzzle. 7 children received a puzzle but not a book and not a toy. Complete the Venn diagram above. [3] (b) There are 40 children in the nursery. Using the Venn diagram, write down and solve an equation in x. Answer(b) [3]

7 0581/41/M/J/14 © UCLES 2014 [Turn over (c) Work out (i) the probability that a child, chosen at random, received a book but not a toy and not a puzzle, Answer(c)(i) … [1] (ii) the number of children who received a book and a puzzle but not a toy, Answer(c)(ii) … [1] (iii) n(B), Answer(c)(iii) … [1] (iv) n(B ∪ P), Answer(c)(iv) … [1] (v) n(B ∪ T ∪ P)'. Answer(c)(v) … [1] (d) B T P Shade the region B ∩ (T ∪ P)'. [1] __________________________________________________________________________________________

8 0581/41/M/J/14 © UCLES 2014 5 North Scale: 2 cm to 3 km P S L In the scale drawing, P is a port, L is a lighthouse and S is a ship. The scale is 2 centimetres represents 3 kilometres. (a) Measure the bearing of S from P. Answer(a) … [1] (b) Find the actual distance of S from L. Answer(b) … km [2] (c) The bearing of L from S is 160°. Calculate the bearing of S from L. Answer(c) … [1]

9 0581/41/M/J/14 © UCLES 2014 [Turn over (d) Work out the scale of the map in the form 1 : n. Answer(d) 1 : … [2] (e) A boat B is ● equidistant from S and L and ● equidistant from the lines PS and SL. On the diagram, using a straight edge and compasses only, construct the position of B. [5] (f) The lighthouse stands on an island of area 1.5 cm2 on the scale drawing. Work out the actual area of the island. Answer(f) … km2 [2] __________________________________________________________________________________________

10 0581/41/M/J/14 © UCLES 2014 6 (a) A square spinner is biased. The probabilities of obtaining the scores 1, 2, 3 and 4 when it is spun are given in the table. Score 1 2 3 4 Probability 0.1 0.2 0.4 0.3 (i) Work out the probability that on one spin the score is 2 or 3. Answer(a)(i) … [2] (ii) In 5000 spins, how many times would you expect to score 4 with this spinner? Answer(a)(ii) … [1] (iii) Work out the probability of scoring 1 on the fi rst spin and 4 on the second spin. Answer(a)(iii) … [2] (b) In a bag there are 7 red discs and 5 blue discs. From the bag a disc is chosen at random and not replaced. A second disc is then chosen at random. Work out the probability that at least one of the discs is red. Give your answer as a fraction. Answer(b) … [3] __________________________________________________________________________________________

11 0581/41/M/J/14 © UCLES 2014 [Turn over 7 A y x 4 3 2 1 –1 –2 –3 –4 –5 0 –1 1 2 3 4 5 6 –2 –3 –4 –5 –6 (a) On the grid, (i) draw the image of shape A after a translation by the vector 4 5 - - e o, [2] (ii) draw the image of shape A after a rotation through 90° clockwise about the origin. [2] (b) (i) On the grid, draw the image of shape A after the transformation represented by the matrix 2 0 0 1 f p. [3] (ii) Describe fully the single transformation represented by the matrix 2 0 0 1 f p. Answer(b)(ii) … … [3] __________________________________________________________________________________________

12 0581/41/M/J/14 © UCLES 2014 8 (a) Complete the table of values for y = x3 – 3x + 1 . x –2.5 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2 2.5 y –7.125 –1 3 1 –0.375 –1 –0.125 3 9.125 [2] (b) Draw the graph of y = x3 – 3x + 1 for –2.5 Ğ x Ğ 2.5 . y x 10 9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 0 –1 –2 –3 3 2 1 [4]

13 0581/41/M/J/14 © UCLES 2014 [Turn over (c) By drawing a suitable tangent, estimate the gradient of the curve at the point where x = 2. Answer(c) … [3] (d) Use your graph to solve the equation x3 – 3x + 1 = 1 . Answer(d) x = … or x = … or x = … [2] (e) Use your graph to complete the inequality in k for which the equation x3 – 3x + 1 = k has three different solutions. Answer(e) … < k < … [2] __________________________________________________________________________________________

14 0581/41/M/J/14 © UCLES 2014 9 80 70 60 50 40 30 20 10 10 20 30 Time (minutes) 40 50 0 Cumulative frequency t The times (t minutes) taken by 80 people to complete a charity swim were recorded. The results are shown in the cumulative frequency diagram above. (a) Find (i) the median, Answer(a)(i) … min [1] (ii) the inter-quartile range, Answer(a)(ii) … min [2]

15 0581/41/M/J/14 © UCLES 2014 [Turn over (iii) the 70th percentile. Answer(a)(iii) … min [2] (b) The times taken by the 80 people are shown in this grouped frequency table. Time (t minutes) 0 < t Ğ 20 20 < t Ğ 30 30 < t Ğ 45 45 < t Ğ 50 Frequency 12 21 33 14 (i) Calculate an estimate of the mean time. Answer(b)(i) … min [4] (ii) Draw a histogram to represent the grouped frequency table. 4 3 2 1 10 20 30 Time (minutes) 40 50 0 Frequency density t [4] __________________________________________________________________________________________

16 0581/41/M/J/14 © UCLES 2014 10 (a) f(x) = 2x – 3 g(x) = 1 1 x + + 2 h(x) = 3x (i) Work out f(4). Answer(a)(i) … [1] (ii) Work out fh(–1). Answer(a)(ii) … [2] (iii) Find f –1(x), the inverse of f(x). Answer(a)(iii) f –1(x) = … [2] (iv) Find ff(x) in its simplest form. Answer(a)(iv) ff(x) = … [2]

17 0581/41/M/J/14 © UCLES 2014 [Turn over (v) Show that the equation f(x) = g(x) simplifi es to 2x2 – 3x – 6 = 0 . Answer(a)(v) [3] (vi) Solve the equation 2x2 – 3x – 6 = 0 . Give your answers correct to 2 decimal places. Show all your working. Answer(a)(vi) x = … or x = … [4] (b) Simplify 2 2 3 10 3 2 x x x x + + - - . Answer(b) … [4] __________________________________________________________________________________________

18 0581/41/M/J/14 © UCLES 2014 11 (a) = 3 4 - e o (i) P is the point (–2, 3). Work out the co-ordinates of Q. Answer(a)(i) (… , …) [1] (ii) Work out  , the magnitude of . Answer(a)(ii) … [2]

19 0581/41/M/J/14 © UCLES 2014 [Turn over (b) A N O Y C B a b NOT TO SCALE OACB is a parallelogram. = a and = b. AN : NB = 2 : 3 and AY = 5 2 AC. (i) Write each of the following in terms of a and/or b. Give your answers in their simplest form. (a) Answer(b)(i)(a) = … [2] (b) Answer(b)(i)(b) = … [2] (ii) Write down two conclusions you can make about the line segments NY and BC. Answer(b)(ii) … … [2] __________________________________________________________________________________________

20 0581/41/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 0581/41 Paper 4 (Extended), maximum raw mark 130 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 41 © 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 Qu Answers Mark Part Marks 1 (a) (i)       − 2 2 4 6 1 (ii) Not possible 1 (iii)       − 2 2 4 6 2 B1 for one row or column correct (iv)       − 3 1 2 1 5 1 oe isw 2 B1 for       d b c a 5 1 seen or       − 3 1 2 1 k seen (b) 1 column in C and 2 rows in D 1 Any clear indication (c) Enlargement [Factor] 2 [Centre] (0, 0) oe 1 1 1 2 (a) 8 2 M1 for 12 ÷ 1.5 oe (b) [Distance =] 36 their36 ÷ 3 [= 12] oe B1 M1 (c) 200 2 M1 for 12 × 1000 ÷ 60 oe e.g. 36 000 ÷ 180 (d) Horizontal line at 36 to 13 45 (their 13 45, 36) joined to (16 42, 0) 1 1FT 3 (a) 62 705 2 M1 for 75 246 ÷ 6 soi by 12 541 or 75 246 × 5 (b) 10.9 or 10.88… 3 M2 for ( ) 890 135 890 135 675 150 − × 100 oe or M1 for correct fraction soi by 0.1088… or 890 135 675 150 × 100 soi by 110.88…

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 41 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks (c) 127 000 3 M2 for 135 890 ÷ 1.07 oe or M1 for 135 890 associated with 107% (d) (i) 59 112 to 59 113 or 59 100 or 59 110 or 59 119 to 59 120 or 59 100 nfww 3 M2 for π × 21 × (302 – 22) oe Or M1 for π × 21 × 302 or π × 21 × 22 (ii) (a) 0.0125 1 (b) 7580 or 7582 or 7581 or 7583 nfww 4 M1 for 21 × 29.7 × their 0.0125 [=7.796 or 7.8[0]] and M1 for their (d)(i) ÷ (21 × 29.7 × their 0.0125) A1 for 7580 to 7583.2 (non integer) If 0 then SC1 for their (d)(i) ÷ (21 × 29.7 × 0.125) 4 (a) 4 – x correctly placed 5 – x correctly placed 7 correctly placed 1 1 1 SC3 for 1, 2 and 7 all correctly placed instead of expressions in x (b) 4 + 11 + (6 – x) + x + 9 + (4 – x) + (5 – x) + 7 = 40 oe 46 – 2x = 40 nfww x = 3 M1 A1 B1 FT from their Venn diagram, condone omission of one subset Must be in the form a + bx = c, ie each side simplified, or better (c) (i) 40 9 or 0.225 or 22.5% 1 ISW cancelling or conversion after correct answer seen (ii) 2 1FT FT from their Venn diagram and their x provided n(B ∩ P ∩ T’) ≠ 5 (iii) 15 1FT FT from their Venn diagram (iv) 25 1FT FT from their Venn diagram (v) 4 1 (d) Correct region shaded. 1 T B P

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 41 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 5 (a) [0]44 to [0]48 1 (b) 12.6 to 13.2 2 B1 for 8.4 to 8.8 seen (c) 340 1 (d) 1 : 150 000 2 M1 for × 100 000 soi (e) Arcs for perp bisector of SL Ruled perp bisector of SL Arcs for bisector of angle PSL Ruled bisector of angle PSL B marked within accuracy 1 1 1 1 1 Two pairs of correct arcs Within tolerance of overlay Marks on PS and SL plus one pair of correct arcs Within tolerance of overlay Within tolerance of overlay Dep on two correct bisectors drawn (f) 3.375 2 M1 for 1.5 × 1.52 or (2/3)2 seen 6 (a) (i) 0.6 oe 2 M1 for 0.2 + 0.4 (ii) 1500 1 (iii) 0.03 oe 2 M1 for 0.1 × 0.3 (b) 132 112 oe 33 28 = 0.848[4…] 3 M2 for 1 – 12 5 ×11 4 or 11 6 12 7 11 7 12 5 11 5 12 7 × + × + × or 11 7 12 5 12 7 × + or M1 for addition of any two of 11 6 12 7 , 11 7 12 5 , 11 5 12 7 × × × or sum of 3 products with an error in the numerator of one product or for 12 5 × 11 4 identified

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 41 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 7 (a) (i) Image: (–4, –3), (–4, –1), (–3, –1) 2 SC1 for translation      − k 5 or       −4 k (ii) Image: (1, –1), (3, –1), (3, –2) 2 SC1 for rotation about the origin but 90º anticlockwise (b) (i) Image: (2, 1), (2, 3), (4, 3) 3 B2 for 2 correct vertices plotted or SC2 for 3 vertices shown in working or SC1 for 2 vertices shown in working or M1       1 0 0 2 ×       3 3 1 2 1 1 (ii) Stretch [factor] 2 Invariant line y-axis oe 1 1 1 Accept x = 0, stays the same 8 (a) 2.125 and 2.375 2 B1 for one correct value (b) Correct curve B4 B3FT for 11 correct plots or B2FT for 9 or 10 correct plots or B1FT for 7 or 8 correct plots (c) Ruled tangent at x = 2 Gradient from 7.8 to 10.2 B1 2 No daylight at x = 2. Consider point of contact as midpoint between two vertices of daylight, this must be between x = 1.8 and 2.2 Dep on B1 awarded Allow integer/integer or a mixed number if within range or M1 dep for (change in y) ÷ (change in x) Dependent on any tangent drawn or close attempt at a tangent at any point Must see correct or implied calculation from a drawn tangent (d) 0 and –1.75 to –1.65 and 1.65 to 1.75 2 B1 for two correct values (e) –1.2 to –0.8 < k < 2.8 to 3.2 2 B1 for each correct or SC1 for reversed answers

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 41 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks 9 (a) (i) 37.5 to 38.5 1 (ii) 19.5 to 20.5 nfww 2 B1 for [LQ =] 23.5 to 24 or [UQ =] 43.5 to 44 (iii) 43 2 B1 for 56 seen or horizontal line drawn at cf = 56 (b) (i) 31.8[4…] nfww 4 M1 for midpoints soi (condone 1 error or omission) and M1 for use of ∑ft with t in correct interval including both boundaries (condone 1 further error or omission) and M1 (dep on 2nd M1) for ∑ft ÷ 80 (2547.5 ÷ 80) (ii) Correct histogram 4 B1 for each correct block with correct width and height If B0 then SC1 for four correct f.d.s or four correct widths 10 (a) (i) 5 1 (ii) – 2 3 1 oe 2 B1 for [h(– 1) =] 3 1 soi or M1 for 2(3x) – 3 (iii) 2 3 + x or 2 x + 1.5 as final ans 2 M1 for y + 3 = 2x or x = 2y – 3 or 2 y = x – 1.5 or better or correct reverse flowchart (iv) 4x – 9 as final answer nfww 2 M1 for 2(2x – 3) – 3 (v) (2x – 3)(x + 1) = 1 + 2(x + 1) 2x2 – 3x + 2x – 3 or better seen 2x2 – 3x – 6 = 0 M1 B1 A1 (2x – 5)(x + 1) = 1 (eliminate fractions) 2x2 – 5x + 2x – 5 or better seen No errors or omissions seen

Page 7 Mark Scheme Syllabus Paper IGCSE – May/June 2014 0581 41 © Cambridge International Examinations 2014 Qu Answers Mark Part Marks (vi) 2 2 6 2 4 ) 3 ( ) 3 ( 2 × − × × − − ± − − 2.64 and – 1.14 cao B2 B1B1 B1 for 6 2 4 ) 3 ( 2 − × × − − or better [√57] and if in form r q p + or r q p − B1 for p = – (– 3) and r = 2 × 2 or better SC1 for 2.64 and –1.14 seen in working or 2.6 and –1.1 as final ans or 2.637. and –1.137.. as final ans or –2.64 and 1.14 as final ans (b) 5 1 + − x x as final answer nfww 4 B3 for (x – 1)(x – 2) and (x + 5)(x – 2) or B2 for (x – 1)(x – 2) or (x + 5)(x – 2) or SC1 for (x + a)(x + b) where a + b = 3 or – 3 or ab = 2 or – 10 11 (a) (i) (– 5, 7) 1 (ii) 5 2 M1 for 2 2 4 ) 3 ( + − or better (b) (i) (a) 5 3 a + 5 2 b or 5 1 (3a + 2b) (a) final answer 2 M1 for any correct vector path for ON (b) 5 2 a 2 M1 for any correct vector path for NY (ii) NY = 5 2 BC oe [NY] parallel to [BC] 1dep 1dep dep on (b)(i)(b) correct dep on NY = ka, k ≠ 1