Cambridge IGCSE Mathematics (with coursework) 0581 — 2013 May/June Paper 4 · Variant 3

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 0581/43 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_43/FP © UCLES 2013 *4906263148* www.XtremePapers.com

2 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 1 (a) Ali and Ben receive a sum of money. They share it in the ratio 5 : 1. Ali receives $2345. Calculate the total amount. Answer(a) $ … [2] (b) Ali uses 11% of his $2345 to buy a television. Calculate the cost of the television. Answer(b) $ … [2] (c) A different television costs $330. (i) Ben buys one in a sale when this cost is reduced by 15%. How much does Ben pay? Answer(c)(i) $ … [2] (ii) $330 is 12% less than the cost last year. Calculate the cost last year. Answer(c)(ii) $ … [3]

3 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (d) Ali invests $1500 of his share in a bank account. The account pays compound interest at a rate of 2.3% per year. Calculate the total amount in the account at the end of 3 years. Answer(d) $ … [3] (e) Ali also buys a computer for $325. He later sells this computer for $250. Calculate Ali’s percentage loss. Answer(e) … % [3] _____________________________________________________________________________________

4 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 2 (a) In this question show all your construction arcs and use only a ruler and compasses to draw the boundaries of your region. This scale drawing shows the positions of four towns, P, Q, R and S, on a map where 1 cm represents 10 km. P S R Q North Scale: 1 cm to 10 km A nature reserve lies in the quadrilateral PQRS. The boundaries of the nature reserve are: ● equidistant from Q and from R ● equidistant from PS and from PQ ● 60 km from R ● along QR. (i) Shade the region which represents the nature reserve. [7] (ii) Measure the bearing of S from P. Answer(a)(ii) … [1]

5 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (b) A circular lake in the nature reserve has a radius of 45 m. (i) Calculate the area of the lake. Answer(b)(i) … m2 [2] (ii) NOT TO SCALE A fence is placed along part of the circumference of the lake. This arc subtends an angle of 210° at the centre of the circle. Calculate the length of the fence. Answer(b)(ii) … m [2] _____________________________________________________________________________________

6 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 3 (a) Luk wants to buy x goats and y sheep. (i) He wants to buy at least 5 goats. Write down an inequality in x to represent this condition. Answer(a)(i) … [1] (ii) He wants to buy at least 11 sheep. Write down an inequality in y to represent this condition. Answer(a)(ii) … [1] (iii) He wants to buy at least 20 animals. Write down an inequality in x and y to represent this condition. Answer(a)(iii) … [1] (b) Goats cost $4 and sheep cost $8. The maximum Luk can spend is $160. Write down an inequality in x and y and show that it simplifi es to x + 2y Y 40 . Answer(b) [1]

7 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) (i) On the grid below, draw four lines to show the four inequalities and shade the unwanted regions. 40 35 30 25 20 15 10 5 y x 0 5 10 15 20 25 30 35 40 [7] (ii) Work out the maximum number of animals that Luk can buy. Answer(c)(ii) … [2] _____________________________________________________________________________________

8 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 4 I G E F J H 40 cm 22 cm 7 cm NOT TO SCALE EFGHIJ is a solid metal prism of length 40 cm. The cross section EFG is a right-angled triangle. EF = 7 cm and EG = 22 cm. (a) Calculate the volume of the prism. Answer(a) … cm3 [2] (b) Calculate the length FJ. Answer(b) FJ = … cm [4]

9 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) Calculate the angle between FJ and the base EGJH of the prism. Answer(c) … [3] (d) The prism is melted and made into spheres. Each sphere has a radius 1.5 cm. Work out the greatest number of spheres that can be made. [The volume, V, of a sphere with radius r is V = 3 4 πr3.] Answer(d) … [3] (e) (i) A right-angled triangle is the cross section of another prism. This triangle has height 4.5 cm and base 11.0 cm. Both measurements are correct to 1 decimal place. Calculate the upper bound for the area of this triangle. Answer(e)(i) … cm2 [2] (ii) Write your answer to part (e)(i) correct to 4 signifi cant fi gures. Answer(e)(ii) … cm2 [1] _____________________________________________________________________________________

10 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 5 (a) Complete this table of values for the function f(x) = x 1 – x2, x ¸ 0. x –3 –2 –1 –0.5 –0.2 0.2 0.5 1 2 3 f(x) –9.33 –4.5 –2 –2.25 4.96 –3.5 –8.67 [3] (b) Draw the graph of f(x) = x 1 – x2 for –3 Y x Y –0.2 and 0.2 Y x Y 3. y x 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 –9 –10 0 1 –1 –2 –3 2 3 [5]

11 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) Use your graph to solve f(x) = –3. Answer(c) x = … or x = … or x = … [3] (d) By drawing a suitable line on your graph, solve the equation f(x) = 2x – 2. Answer(d) x = … or x = … or x = … [3] (e) By drawing a suitable tangent, work out an estimate of the gradient of the curve at the point where x = –2. You must show your working. Answer(e) … [3] _____________________________________________________________________________________

12 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 6 In a box there are 7 red cards and 3 blue cards. A card is drawn at random from the box and is not replaced. A second card is then drawn at random from the box. (a) Complete this tree diagram. Red Blue Red Blue Blue Red … … … … … First card Second card 7 10 [3] (b) Work out the probability that the two cards are of different colours. Give your answer as a fraction. Answer(b) … [3] _____________________________________________________________________________________

13 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use 7 A B x y 10 9 8 7 6 5 4 3 2 1 0 1 2 3 4 5 6 7 8 (a) (i) Draw the image of shape A after a stretch, factor 3, x-axis invariant. [2] (ii) Write down the matrix representing a stretch, factor 3, x-axis invariant. Answer(a)(ii) e o [2] (b) (i) Describe fully the single transformation which maps shape A onto shape B. Answer(b)(i) … [3] (ii) Write down the matrix representing the transformation which maps shape A onto shape B. Answer(b)(ii) e o [2] _____________________________________________________________________________________

14 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 8 (a) 27° B C O A E D NOT TO SCALE A, B, C, D and E are points on the circle centre O. Angle ABD = 27°. Find (i) angle ACD, Answer(a)(i) Angle ACD = … [1] (ii) angle AOD, Answer(a)(ii) Angle AOD = … [1] (iii) angle AED. Answer(a)(iii) Angle AED = … [1] (b) L K N M 67° 100° 45 cm 32 cm NOT TO SCALE The diagram shows quadrilateral KLMN. KL = 45 cm, LN = 32 cm, angle KLN = 100° and angle NLM = 67°.

15 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (i) Calculate the length KN. Answer(b)(i) KN = … cm [4] (ii) The area of triangle LMN is 324 cm2. Calculate the length LM. Answer(b)(ii) LM = … cm [3] (iii) Another triangle XYZ is mathematically similar to triangle LMN. L N M X Z Y NOT TO SCALE XZ = 16 cm and the area of triangle LMN is 324 cm2. Calculate the area of triangle XYZ. Answer(b)(iii) … cm2 [2] _____________________________________________________________________________________

16 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 9 Sam asked 80 people how many minutes their journey to work took on one day. The cumulative frequency diagram shows the times taken (m minutes). 80 70 60 50 40 30 20 10 m 0 10 20 30 40 50 Cumulative frequency Time (minutes) (a) Find (i) the median, Answer(a)(i) … min [1] (ii) the lower quartile, Answer(a)(ii) … min [1] (iii) the inter-quartile range. Answer(a)(iii) … min [1]

17 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (b) One of the 80 people is chosen at random. Find the probability that their journey to work took more than 35 minutes. Give your answer as a fraction. Answer(b) … [2] (c) Use the cumulative frequency diagram to complete this frequency table. Time (m minutes) 0 < m Y 10 10 < m Y 15 15 < m Y 30 30 < m Y 40 40 < m Y 50 Frequency 30 12 18 [2] (d) Using mid-interval values, calculate an estimate of the mean journey time for the 80 people. Answer(d) … min [3] (e) Use the table in part (c) to complete the histogram to show the times taken by the 80 people. One column has already been completed for you. Frequency density Time (minutes) 4 3 2 1 0 10 20 30 40 50 m [5] _____________________________________________________________________________________

18 0581/43/M/J/13 © UCLES 2013 For Examiner′s Use 10 (a) (i) Solve 2(3x – 7) = 13. Answer(a)(i) x = … [3] (ii) Solve by factorising x2 – 7x + 6 = 0. Answer(a)(ii) x = … or x = … [3] (iii) Solve 5 x 3 2 - + 10 2 x + = 4. Answer(a)(iii) x = … [4]

19 0581/43/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (b) 12 = 1 12 + 22 = 5 12 + 22 + 32 = 14 12 + 22 + 32 + 42 = 30 12 + 22 + 32 + 42 + … + n2 = an3 + bn2 + 6 n Work out the values of a and b. Answer(b) a = … b = … [6] _____________________________________________________________________________________

20 0581/43/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 0581/43 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 43 © 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 1 (a) 2814 final answer 2 M1 for 2345 ÷ 5 soi by 469 or ans = 2810 (b) 257.95 final answer 2 M1 for 2345 × 0.11 oe or ans = 258 (c) (i) 280.5[0] final answer 2 M1 for 330 × (1 – 0.15) oe or ans = 281 (ii) 375 3 M2 for 330 ÷ (1 – 0.12) oe Or M1 for 330 = (100 – 12)% oe (d) 1605.89 or 1605.9[0] 3 M2 for 1500 × (1 + 0.023)3 oe soi by 1605.898751 or 1500 × 1.07(05…) Or M1 for 1500 × (1 + 0.023) 2 oe (e) 23.1 or 23.07 to 23.08 3 M2 for 325 250 325 − × 100 oe Or M1 for 325 250 325 − soi by 0.2307… 3sf or better or 325 250 × 100 soi by 76.9…

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 43 © Cambridge International Examinations 2013 2 (a) (i) Perpendicular bisector of QR ruled with 2 correct sets of arcs centred Q and R Bisector of angle SPQ ruled with correct arcs. (Marks on PS and PQ and correct pair of arcs) Compass drawn arc centre R with radius 6 cm (±2 mm) Correct region shaded cao 2 2 B2 1dep B1 for correct bisector ruled B1 for correct angle bisector ruled B1 for any compass drawn arc centre R not used in any construction with no feathering Dependent on all B4 marks for the correct loci (ii) 217 to 221 1 (b) (i) 6360 or 6361 to 6363 2 M1 for π × 452 (ii) 165 or 164.9 to 165 2 M1 for 360 210 × 2π × 45 3 (a) (i) x ≥ 5 1 –1 once for strict inequalities in (i) to (iii) (ii) y ≥ 11 1 (iii) x + y ≥ 20 1 (b) 4x + 8y 160 ≤ and divide by 4 1 If there is a final inequality it must be the given one (c) (i) x = 5 ruled y = 11 ruled x + y = 20 ruled x + 2y = 40 ruled Correct shading of unwanted region 1 1 2 2 1dep Must be on correct grid line Must be on correct grid line B1 for one axis intercept correct when extended if necessary but not parallel to an axis B1 for one axis intercept correct when extended if necessary but not parallel to an axis Dependent on 6 marks earned for the boundaries (ii) 29 2 M1 for x + y evaluated where (x, y) is a point in their quadrilateral and x and y are integers

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 43 © Cambridge International Examinations 2013 4 (a) 3080 2 M1 for ½ × 7 × 22 × 40 (b) 46.2 or 46.18 to 46.2 www 4 M3 for 2 2 2 40 + 22 + 7 or M2 for 72 + 222 + 402 soi by 2133 or M1 for correct Pythagoras on one face (c) 8.7 or 8.7 to 8.72 www 3 M2 for sin– 1 their(b) 7 oe or M1 for sin = their(b) 7 oe (d) 217 3 M1 for 3 4 ×π×1.53 soi by 14.1 to 14.14 and M1 dep for their (a) ÷ their 14.14 soi by 218. Dependent on M1 earned (e) (i) 25.13875 final answer 2 B1 for 4.55 and 11.05 seen or 25.13875 seen and then spoiled (ii) 25.14 1FT Strict FT their (e)(i) correct to 4s.f. if rounding is possible 5 (a) –5.04, 1.75, 0 3 B1 for each correct value (b) Fully correct curve 5 B3FT for 10 correct plots from their (a) B2FT for 8 or 9 correct plots or B1FT for 6 or 7 correct plots and SC1 for two branches not joined (c) –1.6 to – 1.5 –0.4 to –0.3 1.8 to 1.9 1 1 1 (d) –2.6 to –2.5 www –0.4 to –0.3 1 1 1 1 After 0 scored, M1 for y = 2x – 2 drawn (e) 3.25 to 4.25 with correct tangent 3 B1 for correct tangent B2 for answer in range dep on close attempt at tangent M1dep for [–] run rise used with values soi from tangent, dep on correct or close attempt at tangent

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 43 © Cambridge International Examinations 2013 6 (a) 10 3 correctly placed 9 6 and 9 3 correctly placed 9 7 and 9 2 correctly placed 1 1 1 Accept 0.3 Accept 0.667 or better and 0.333 or better Accept 0.778 or better and 0.222 or better (b) 90 42 or 45 21 or 30 14 or 15 7 3 M2 for 10 7 × 9 3 + 10 3 × 9 7 soi by 0.467 or better or M1 for 10 7 × 9 3 or 10 3 × 9 7 soi by 0.233 or better 7 (a) (i) Triangle at (1, 3) (1, 9) (3, 3) 2 SC1 for correct vertices not joined or triangle(1, 1) (3, 1) (1, 7) (ii)       3 0 0 1 2 SC1 for       k 0 0 1 , k ≠ ± 1 or 0 or       1 0 0 3 (b) (i) Shear x-axis oe invariant [factor] 2 1 1 1 (ii)       1 0 2 1 2FT FT from their 2 in (b)(i) SC1 for       1 0 1 k , k ≠ 0 or       1 FT 2 0 1 8 (a) (i) 27 1 (ii) 54 1 (iii) 153 1 (b) (i) 59.6 or 59.57… www 4 M2 for 452 + 322 – 2 × 45 × 32 × cos100 or M1 for implicit cos rule and A1 for 3549…. (ii) 22.[0] or 21.99… www 3 M2 for 324 ÷ (½ × 32 × sin67) or M1 for [324 =] ½ × 32 × x × sin67 (iii) 81[.0] 2 B1 for 22 or (½)2 oe seen or ½ × 16 × ½ their(b)(ii) × sin67

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 43 © Cambridge International Examinations 2013 9 (a) (i) 14 1 (ii) 8 1 (iii) 30 – their (ii) 1FT (b) 80 11 2 SC1 for 80 69 (c) 16, 4 2 B1 for each correct value (d) 18.0625 rot to 3sf or better or 18.1 www 3 M1 for Σmf for m as mid values of 5, 12.5, 22.5, 35 and 45 (= 1445) and M1 dep for Σmf ÷ 80, dep on M1 earned (e) Correct widths with no gaps 2nd block w = 5, fd = 2.4 3rd block w = 15 fd = 1.2 4th block w = 10 and fd = 1.6 5th block w = 10 and fd = 0.4 1 1 1 1FT 1FT Strict FT from their (c) Strict FT from their (c) After 0 scored for blocks, SC1 for 4 correct fds soi by correct heights 10 (a) (i) 4.5 or 4½ 3 M2 for a complete correct method or M1 for one correct step at any stage. (ii) (x – 6)(x – 1) 1, 6 M2 A1FT M1 for (x +a)(x +b) where ab = 6 or a + b = - 7 FT their brackets dep on M1 earned After M0 scored SC1 for 1, 6 as answer (iii) 6 4 B1 for 2(3x – 2) + x + 2 = 4×10 oe and B1 for correct multiplication of a bracket and M1 for correct rearrangement of their linear equation without brackets to ax = b + c + d or better (b) a = 1/3 oe, b = 1/2 oe 6 B1 for any one of 1 = a + b + 1/6 oe 5 = 8a + 4b + 2/6 oe 14 = 27a + 9b + 3/6 oe 30 = 64a + 16b + 4/6 oe Or any other correct equation and B1 for another of the above equations and M1 for equating one coefficient or correct rearrangement to give a or b as subject and M1 for subtracting to eliminate a or b or correct substitution for their a or their b A1 for a = 1/3 oe or b = 1/2 oe