Cambridge IGCSE Mathematics (with coursework) 0581 — 2013 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 a pencil for any diagrams or graphs. Do not use staples, paper clips, highlighters, 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 2013 2 hours 30 minutes Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Tracing paper (optional) UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certifi cate of Secondary Education This document consists of 19 printed pages and 1 blank page. [Turn over IB13 06_0581_41/FP © UCLES 2013 *2690565884* www.XtremePapers.com

2 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 1 (a) One day, Maria took 27 minutes to walk 1.8 km to school. She left home at 07 48. (i) Write down the time Maria arrived at school. Answer(a)(i) … [1] (ii) Show that Maria’s average walking speed was 4 km/h. Answer(a)(ii) [2] (b) Another day, Maria cycled the 1.8 km to school at an average speed of 15 km/h. (i) Calculate the percentage increase that 15 km/h is on Maria’s walking speed of 4 km/h. Answer(b)(i) … % [3] (ii) Calculate the percentage decrease that Maria’s cycling time is on her walking time of 27 minutes. Answer(b)(ii) … % [3]

3 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (iii) After school, Maria cycled to her friend’s home. This took 9 minutes, which was 36% of the time Maria takes to walk to her friend’s home. Calculate the time Maria takes to walk to her friend’s home. Answer(b)(iii) … min [2] _____________________________________________________________________________________

4 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 2 f(x) = 3 – x – x2 g(x) = 3x (a) Complete the tables of values for f(x) and g(x). x –1.5 –1 –0.5 0 0.5 1 1.5 f(x) 2.25 3 3.25 2.25 1 –0.75 x –1.5 –1 –0.5 0 0.5 1 1.5 g(x) 0.19 0.58 1.73 3 5.20 [3] (b) On the grid, draw the graphs of y = f(x) and y = g(x) for –1.5 Ğ x Ğ 1.5 . y x 6 5 4 3 2 1 –1 –2 0 0.5 1 1.5 –0.5 –1 –1.5 [6]

5 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) For –1.5 Ğ x Ğ 1.5, use your graphs to solve (i) f(x) = 0, Answer(c)(i) x = … [1] (ii) g(x) = 4, Answer(c)(ii) x = … [1] (iii) f(x) = g(x). Answer(c)(iii) x = … [1] (d) By drawing a suitable tangent, fi nd an estimate of the gradient of the graph of y = f(x) when x = 0.5. Answer(d) … [3] _____________________________________________________________________________________

6 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 3 200 students estimate the mass (m grams) of a coin. The cumulative frequency diagram shows the results. 200 180 160 140 120 100 80 60 40 20 2 4 6 Mass (grams) 8 10 1 0 3 5 7 9 Cumulative frequency m

7 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (a) Find (i) the median, Answer(a)(i) … g [1] (ii) the upper quartile, Answer(a)(ii) … g [1] (iii) the 80th percentile, Answer(a)(iii) … g [1] (iv) the number of students whose estimate is 7 g or less. Answer(a)(iv) … [1] (b) (i) Use the cumulative frequency diagram to complete the frequency table. Mass (m grams) 0 < m Ğ 2 2 < m Ğ 4 4 < m Ğ 6 6 < m Ğ 8 8 < m Ğ 10 Frequency 40 2 [2] (ii) A student is chosen at random. The probability that the student estimates that the mass is greater than M grams is 0.3. Find the value of M. Answer(b)(ii) M = … [2] _____________________________________________________________________________________

8 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 4 Q y x 9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –1 1 0 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 –8 R (a) Describe fully the single transformation that maps shape Q onto shape R. Answer(a) … [3] (b) (i) Draw the image when shape Q is translated by the vector 4 5 e o. [2] (ii) Draw the image when shape Q is refl ected in the line x = 2. [2] (iii) Draw the image when shape Q is stretched, factor 3, x-axis invariant. [2] (iv) Find the 2 × 2 matrix that represents a stretch of factor 3, x-axis invariant. Answer(b)(iv) e o [2] (c) Describe fully the single transformation represented by the matrix 0 1 1 0 e o. Answer(c) … [2] _____________________________________________________________________________________

9 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use 5 Height (h cm) 150 < h Ğ 160 160 < h Ğ 165 165 < h Ğ 180 180 < h Ğ 190 Frequency 5 9 18 10 The table shows information about the heights of a group of 42 students. (a) Using mid-interval values, calculate an estimate of the mean height of the students. Show your working. Answer(a) … cm [3] (b) Write down the interval which contains the lower quartile. Answer(b) … [1] (c) Complete the histogram to show the information in the table. One column has already been drawn for you. 150 155 160 165 170 175 180 185 190 2 1 0 Frequency density Height (cm) [4] _____________________________________________________________________________________

10 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 6 A E B C D 52° 30° 15.7 cm 23.4 cm 16.5 cm NOT TO SCALE In the diagram, BCD is a straight line and ABDE is a quadrilateral. Angle BAC = 90°, angle ABC = 30° and angle CAE = 52°. AC = 15.7 cm, CE = 16.5 cm and CD = 23.4 cm. (a) Calculate BC. Answer(a) BC = … cm [3] (b) Use the sine rule to calculate angle AEC. Show that it rounds to 48.57°, correct to 2 decimal places. Answer(b) [3]

11 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) (i) Show that angle ECD = 40.6°, correct to 1 decimal place. Answer(c)(i) [2] (ii) Calculate DE. Answer(c)(ii) DE = … cm [4] (d) Calculate the area of the quadrilateral ABDE. Answer(d) … cm2 [4] _____________________________________________________________________________________

12 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 7 (a) A B C (2x + 3) cm (x + 2) cm NOT TO SCALE In triangle ABC, AB = (x + 2) cm and AC = (2x + 3) cm. sin ACB = 16 9 Find the length of BC. Answer(a) BC = … cm [6] (b) A bag contains 7 white beads and 5 red beads. (i) The mass of a red bead is 2.5 grams more than the mass of a white bead. The total mass of all the 12 beads is 114.5 grams. Find the mass of a white bead and the mass of a red bead. Answer(b)(i) White … g Red … g [5]

13 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (ii) Two beads are taken out of the bag at random, without replacement. Find the probability that (a) they are both white, Answer(b)(ii)(a) … [2] (b) one is white and one is red. Answer(b)(ii)(b) … [3] _____________________________________________________________________________________

14 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 8 (a) D C E A B 84° 110° 110° x° x° NOT TO SCALE In the pentagon ABCDE, angle EAB = angle ABC = 110° and angle CDE = 84°. Angle BCD = angle DEA = x°. (i) Calculate the value of x. Answer(a)(i) x = … [2] (ii) BC = CD. Calculate angle CBD. Answer(a)(ii) Angle CBD = … [1] (iii) This pentagon also has one line of symmetry. Calculate angle ADB. Answer(a)(iii) Angle ADB = … [1] (b) A, B and C lie on a circle centre O. Angle AOC = 3y° and angle ABC = (4y + 4)°. Find the value of y. Answer(b) y = … [4] O A B C 3y° (4y + 4)° NOT TO SCALE

15 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) S R Q P 78° NOT TO SCALE In the cyclic quadrilateral PQRS, angle SPQ = 78°. (i) Write down the geometrical reason why angle QRS = 102°. Answer(c)(i) … [1] (ii) Angle PRQ : Angle PRS = 1 : 2. Calculate angle PQS. Answer(c)(ii) Angle PQS = … [3] (d) l cm 5 cm2 6.9 cm 7.2 cm2 NOT TO SCALE The diagram shows two similar fi gures. The areas of the fi gures are 5 cm2 and 7.2 cm2. The lengths of the bases are l cm and 6.9 cm. Calculate the value of l. Answer(d) l = … [3] _____________________________________________________________________________________

16 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 9 f(x) = x2 + x – 3 g(x) = 2x + 7 h(x) = 2x (a) Solve the equation f(x) = 0. Show all your working and give your answers correct to 2 decimal places. Answer(a) x = … or x = … [4] (b) fg(x) = px2 + qx + r Find the values of p, q and r. Answer(b) p = … q = … r = … [3]

17 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) Find g –1(x). Answer(c) g –1(x) = … [2] (d) Find x when h(x) = 0.25. Answer(d) x = … [1] (e) Find hhh(3). Give your answer in standard form, correct to 4 signifi cant fi gures. Answer(e) … [4] _____________________________________________________________________________________

18 0581/41/M/J/13 © UCLES 2013 For Examiner′s Use 10 Star 3 Star 2 Star 1 The diagrams show a sequence of stars made of lines and dots. (a) Complete the table for Star 5, Star 7 and Star n. Star 1 Star 2 Star 3 Star 4 Star 5 Star 7 Star n Number of lines 10 20 30 40 Number of dots 11 21 31 41 [4] (b) The sums of the number of dots in two consecutive stars are shown in the table. Star 1 and Star 2 Star 2 and Star 3 Star 3 and Star 4 32 52 72 Find the sum of the number of dots in (i) Star 10 and Star 11, Answer(b)(i) … [1] (ii) Star n and Star (n + 1), Answer(b)(ii) … [1] (iii) Star (n + 7) and Star (n + 8). Answer(b)(iii) … [1]

19 0581/41/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) The total number of dots in the fi rst n stars is given by the expression 5n2 + 6n . (i) Show that this expression is correct when n = 3. Answer(c)(i) [2] (ii) Find the total number of dots in the fi rst 10 stars. Answer(c)(ii) … [1] (d) The total number of dots in the fi rst n stars is 5n2 + 6n . The number of dots in the (n + 1)th star is 10(n + 1) + 1. Add these two expressions to show that the total number of dots in the fi rst (n + 1) stars is 5(n + 1)2 + 6(n + 1) . You must show each step of your working. Answer(d) [4] _____________________________________________________________________________________

20 0581/41/M/J/13 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. BLANK PAGE © UCLES 2013

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2013 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 2013 series for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level components and some Ordinary Level components. www.XtremePapers.com

Page 2 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 41 © Cambridge International Examinations 2013 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. Answer Mark Part marks 1 (a) (i) [0]8 15 1 (ii) 27 8.1 × 60 [= 4] oe M2 M1 for 27 8.1 oe [0.0667 or better] (b) (i) 275 3 M2 for 4 4 15 − × 100 or 4 15 × 100 – 100 oe or M1 for 4 4 15 − or 4 15 × 100 or oe 375 (ii) 73.3[3…] 3 M2 for 15 8.1 × 60 [=7.2 min] and 27 2.7 27 their − × 100 oe or M1 for 15 8.1 × 60 [=7.2 min] or final answer of 26.6[6…] or 26.7 (iii) 25 2 M1 for 36 9 figs oe

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 41 © Cambridge International Examinations 2013 Qu. Answer Mark Part marks 2 (a) 3, 0.33[3…], 1 3 B1 for each correct value (b) Correct quadratic curve Correct exponential curve 3 3 B2FT for 7 correct points or B1FT for 5 or 6 correct points B2FT for 7 correct points or B1FT for 5 or 6 correct points (c) (i) Answer in range 1.2 < x < 1.4 1 (ii) Answer in range 1.2 < x < 1.35 1 Not from a line other than y = 4 (±1mm) (iii) Answer in range 0.55 < x < 0.7 1 (d) Correct tangent drawn And answer in range –2.5 < m < –1.5 3 B1 for correct tangent at x = 0.5 B2 for answer in range dep on close attempt at tangent M1 for [–] run rise used with values soi from tangent, dep on close attempt at tangent or answer in range 1.5 < m < 2.5 or SC1 for close attempt at tangent to exponential curve and answer in the range 1.6 < m < 2.2 3 (a) (i) 3.2 1 (ii) 4.2 1 (iii) 4.6 1 (iv) 196 1 (b) (i) 100, 46, 12 2 B1 for 2 correct (ii) 4 2 M1 for frequency of 60 or 140 seen in workspace 1.5 1 0.5 0 -0.5 -1 .5 6 4 2 0 -2 1.5 1 0.5 0 -0.5 -1 .5 6 4 2 0 -2 1.5 1 0.5 0 -0.5 -1 .5 6 4 2 0 -2 1.5 1 0.5 0 -0.5 -1 .5 6 4 2 0 -2

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 41 © Cambridge International Examinations 2013 Qu. Answer Marks Part marks 4 (a) Enlargement [centre] (–3, 4) [scale factor] 3 1 1 1 Do not allow column vector for coordinates (b) (i) Image at (1 5), (4, 5), (4, 6), (1, 7) 2 SC1 for translation by       k 5 or       4 k (ii) Image at (5, 1), (8, 1), (8, 3), (5, 2) 2 SC1 for reflection in y = 2 (iii) Image at (– 4, 3), (– 1, 3), (– 1, 6), (– 4, 9) 2 SC1 for three correct vertices or shape with vertices at (–4, 1) and (–1, 1), (–1, 4) and (–4, 7) (iv)       3 0 0 1 2 SC1 for       k 0 0 1 , k ≠ ± 1 or       1 0 0 3 (c) Reflection y = x oe 2 B1 B1 independent 5 (a) 171.25 (or 171 or 171.2 or 171.3) www 3 M1 for 5 × 155 + 9 × 162.5 + 18 × 172.5 + 10 × 185 [= 7192.5] and M1 (dep on M1) for their Σfx ÷ 42 (b) 160 < x ≤ 165 oe 1 (c) Blocks with heights of 1.8, 1.2, 1, with correct interval widths and no gaps 4 B3 for 2 correct blocks or B2 for 1 correct block or B1 for 3 correct frequency densities or heights or 3 correct widths

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 41 © Cambridge International Examinations 2013 Qu. Answer Marks Part marks 6 (a) 31.4 3 M2 for 30 sin 7. 15 or M1 for correct implicit statement (b) [sinE =] 5. 16 52 sin 7. 15 × 48.573… M2 A1 M1 for correct implicit statement (c) (i) [∠ACE = ] 180 – 52 – 48.57 [= 79.43] [∠ECD = ] 40.57… M1 A1 (ii) 15.3 or 15.27 to 15.281 www 4 M2 for [(DE)2 =] 16.52 + 23.42 – 2 × 16.5 × 23.4cos(40.6 or 40.57) or M1 for full correct implicit statement A1 for 233 to 234 (d) 466 or 466.34 to 466.5 4 M1 for 0.5 × 15.7 × their 31.4 sin(90 – 30) oe M1 for 0.5 × 15.7 × 16.5 sin(128 – their 48.6 or 48.57) oe M1 for 0.5 × 16.5 × 23.4 sin (40.6 or 40.57) oe

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 41 © Cambridge International Examinations 2013 Qu. Answer Mark Part marks 7 (a) 6.61 (6.614…) www 6 B1 for 16 9 3 2 2 = + + x x oe M1 for 16(x + 2) = 9(2x + 3) or better A1 for [x =] 2.5 M2 for √{(2 × their x + 3)2 – (their x + 2)2} or M1 for (2 × their x + 3)2 – (their x + 2)2 or SC2 for final answer of 4√13 or 2 15 7 or better SC1 for final answer of 5√7 or better (b) (i) White = 8.5, red = 11 5 B3 for 7w + 5(w + 2.5) = 114.5 or for 5. 114 5 ) 5.2 ( 7 = + − r r oe B1 for 8.5 or 11 or SC2 for 7w + 5 × w + 2.5 = 114.5 leading to 9.33[3…] or SC1 for 7w + 5 × w + 2.5 = 114.5 OR B1 for r = w + 2.5 oe B1 for 7w + 5r = 114.5 oe M1 for elimination of a variable A1 for 8.5 or 11 (ii) (a) 132 42 or 66 21 or 44 14 or 22 7 (0.318 or 0.3181 to 0.3182) 2 M1 for 12 7 × 11 6 (ii) (b) 132 70 or 66 35 (0.53[0] or 0.5303…) 3 M2 for 12 7 × 11 5 + 12 5 × 11 7 or 1 – their (a) – 12 5 × 11 4 or M1 for 12 7 × 11 5 or 132 35 or SC1 for 144 70 oe from replacement

Page 7 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 41 © Cambridge International Examinations 2013 Qu. Answer Mark Part marks 8 (a) (i) 118 2 M1 for (3 × 180 – 2 × 110 – 84) [÷ 2] or better (ii) 31 1FT FT (180 – their (i)) ÷ 2 (iii) 22 1FT FT 84 – 2 × their (ii) or 2 × their (ii) – 40, only if positive answer and less than 84 (b) 32 4 B2 for 360 – 3y = 2(4y + 4) oe and B1 for 11y = 352 oe or M1 for angle at centre = 2 × angle at circumference soi (c) (i) Opposite angles [cyclic quad] add to 180 1 (ii) 68 3 M1 for [angle PRS =] 102 ÷ 3 × 2 and M1 for angle PQS = angle PRS or angle PRQ = angle PSQ (d) 5.75 3 M2 for 6.9 × 2.7 5 oe or M1 for evidence of ratio of areas = (ratio of sides)2 or sf = 1.2 9 (a) 2 ) 3 ( 1 4 1 1 2 − × × − ± − –2.30, 1.30 final answer 2 2 B1 for ) 3 ( 1 4 12 − × × − or better and if in the form r q p + or r q p − then B1 for p = –1 and r = 2(1) or better B1 B1 SC1 for –2.30 and 1.30 seen or –2.3 or –2.303 to – 2.302 and 1.3 or 1.302 to 1.303 or final answer –1.30 and 2.30 (b) 4, 30, 53 3 M1 for (2x + 7)2 + (2x + 7) – 3 and B1 for (2x + 7)2 = 4x2 + 14x + 14x + 49 oe

Page 8 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 41 © Cambridge International Examinations 2013 Qu. Answer Mark Part marks (c) 2 7 − x 2 M1 for y – 7 = 2x or x = 2y + 7 or –7 then ÷ 2 clearly seen in correct order with arrow or better or 2 7 − y (d) –2 1 (e) 1.158 × 1077 4 B3 for 1.16 × 1077 or 1.1579… × 1077 or 1.157 × 1077 or B2 for 2256 seen or B1 for 28 seen or 256 10 (a) 50, 70 10n oe 51, 71 10n + 1 oe 1 1 1 1 (b) (i) 212 1 (ii) 20n + 12 1 (iii) 20n + 152 1 (c) (i) 5 × 32 + 6 × 3 = 63 and 11 + 21 + 31 = 63 or 32 + 31 = 63 or 11 + 52 = 63 1 1 (ii) 560 1 (d) Complete solution with no errors seen and a conclusion e.g. 5n2 + 6n + 10(n + 1) + 1 = 5n2 + 6n + 10n + 10 + 1 = 5n2 + 10n + 5 + 6n + 6 = 5n2 + 10n + 5 + 6n + 6 = 5(n + 1)2 + 6(n + 1) 4 B1 for 5n2 + 6n + 10n + 10 + 1 or better B1 for use of 5(n + 1)2 = 5n2 + 10n + 5 oe at any stage B1 for use of 6n + 6 = 6(n + 1) oe at any stage