Cambridge IGCSE Mathematics (with coursework) 0581 — 2013 May/June Paper 4 · Variant 2

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 0581/42 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_42/FP © UCLES 2013 *7410003596* www.XtremePapers.com

2 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 1 A tennis club has 560 members. (a) The ratio men : women : children = 5 : 6 : 3. (i) Show that the club has 240 women members. Answer(a)(i) [2] (ii) How many members are children? Answer(a)(ii) … [1] (b) 8 5 of the 240 women members play in a tournament. How many women members do not play in the tournament? Answer(b) … [2] (c) The annual membership fee in 2013 is $198 for each adult and $75 for each child. (i) Calculate the total amount the 560 members pay in 2013. Answer(c)(i) $ … [2] (ii) The adult fee of $198 in 2013 is 5.6% more than the fee in 2012. Calculate the adult fee in 2012. Answer(c)(ii) $ … [3]

3 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (d) The club buys 36 tennis balls for $9.50 and sells them to members for $0.75 each. Calculate the percentage profi t the club makes. Answer(d) … % [3] (e) A tennis court is a rectangle with length 23.7 m and width 10.9 m, each correct to 1 decimal place. Calculate the upper and lower bounds of the perimeter of the court. Answer(e) Upper bound … m Lower bound … m [3] _____________________________________________________________________________________

4 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 2 (a) Q P y x 6 5 4 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 0 –1 1 2 3 4 5 6 7 8 –2 –3 –4 –5 –6 –7 (i) Describe fully the single transformation which maps shape P onto shape Q. Answer(a)(i) … [2] (ii) On the grid above, draw the image of shape P after refl ection in the line y = –1. [2] (iii) On the grid above, draw the image of shape P under the transformation represented by the matrix 1 0 1 0 - e o. [3]

5 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (b) M L y x 10 9 8 7 6 5 4 3 2 1 –1 –2 –3 –4 –5 0 –1 1 2 3 4 5 6 7 8 9 10 11 12 –2 –3 –4 (i) Describe fully the single transformation which maps shape M onto shape L. Answer(b)(i) … [3] (ii) On the grid above, draw the image of shape M after enlargement by scale factor 2, centre (5, 0). [2] _____________________________________________________________________________________

6 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 3 The table shows some values for the function y = 11x – 2x2 – 12 for 1 Ğ x Ğ 4.5. x 1 1.5 2 2.5 3 3.5 4 4.5 y –3 2 3 3 (a) Complete the table of values. [3] (b) On the grid below, draw the graph of y = 11x – 2x2 – 12 for 1 Ğ x Ğ 4.5. y 4 3 2 1 0 –1 –2 –3 0.5 1 1.5 2 2.5 3 4 3.5 4.5 x [4]

7 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) By drawing a suitable line, use your graph to solve the equation 11x – 2x2 = 11. Answer(c) x = … or x = … [2] (d) The line y = mx + 2 is a tangent to the curve y = 11x – 2x2 – 12 at the point P. By drawing this tangent, (i) fi nd the co-ordinates of the point P, Answer(d)(i) (… , …) [2] (ii) work out the value of m. Answer(d)(ii) m = … [2] _____________________________________________________________________________________

8 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 4 A M O N C B T NOT TO SCALE A, B and C lie on the circle centre O, radius 8.5 cm. AB = BC = 10.7 cm. OM is perpendicular to AB and ON is perpendicular to BC. (a) Calculate the area of the circle. Answer(a) … cm2 [2] (b) Write down the length of MB. Answer(b) … cm [1] (c) Calculate angle MOB and show that it rounds to 39° correct to the nearest degree. Answer(c) [2] (d) Using angle MOB = 39°, calculate the length of the major arc AC. Answer(d) … cm [3] (e) The tangents to the circle at A and at C meet at T. Explain clearly why triangle ATB is congruent to triangle CTB. Answer(e) [3] _____________________________________________________________________________________

9 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use 5 Paul buys a number of large sacks of fertiliser costing $x each. He spends $27. (a) Write down, in terms of x, an expression for the number of large sacks which Paul buys. Answer(a) … [1] (b) Rula buys a number of small sacks of fertiliser. Each small sack costs $2 less than a large sack. Rula spends $25. Write down, in terms of x, an expression for the number of small sacks which Rula buys. Answer(b) … [1] (c) Rula buys 4 more sacks than Paul. Write down an equation in x and show that it simplifi es to 2x2 – 3x – 27 = 0. Answer(c) [4] (d) Solve 2x2 – 3x – 27 = 0. Answer(d) x = … or x = … [3] (e) Calculate the number of sacks which Paul buys. Answer(e) … [1] _____________________________________________________________________________________

10 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 6 (a) 15 cm 21 cm 12 cm L M N NOT TO SCALE The diagram shows triangle LMN with LM = 12 cm, LN = 15 cm and MN = 21 cm. (i) Calculate angle LMN. Show that this rounds to 44.4°, correct to 1 decimal place. Answer(a)(i) [4] (ii) Calculate the area of triangle LMN. Answer(a)(ii) … cm2 [2]

11 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (b) 43° 82° 6.4 cm P Q R NOT TO SCALE The diagram shows triangle PQR with PQ = 6.4 cm, angle PQR = 82° and angle QPR = 43°. Calculate the length of PR. Answer(b) PR = … cm [4] _____________________________________________________________________________________

12 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 7 A = 5 7 e o B = (6 – 4) C = 2 1 4 3 e o D = 1 3 - 2 9 - e o (a) Calculate the result of each of the following, if possible. If a calculation is not possible, write “not possible” in the answer space. (i) 3A Answer(a)(i) [1] (ii) AC Answer(a)(ii) [1] (iii) BA Answer(a)(iii) [2] (iv) C + D Answer(a)(iv) [1] (v) D2 Answer(a)(v) [2] (b) Calculate C–1, the inverse of C. Answer(b) [2] _____________________________________________________________________________________

13 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use 8 In this question, give all your answers as fractions. When Ivan goes to school in winter, the probability that he wears a hat is 8 5 . If he wears a hat, the probability that he wears a scarf is 3 2 . If he does not wear a hat, the probability that he wears a scarf is 6 1 . (a) Complete the tree diagram. Hat No hat Scarf No scarf No scarf Scarf … … … … … … [3] (b) Find the probability that Ivan (i) does not wear a hat and does not wear a scarf, Answer(b)(i) … [2] (ii) wears a hat but does not wear a scarf, Answer(b)(ii) … [2] (iii) wears a hat or a scarf but not both. Answer(b)(iii) … [2] (c) If Ivan wears a hat and a scarf, the probability that he wears gloves is 10 7 . Calculate the probability that Ivan does not wear all three of hat, scarf and gloves. Answer(c) … [3] _____________________________________________________________________________________

14 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 9 (a) 12 cm 4 cm NOT TO SCALE The diagram shows a prism of length 12 cm. The cross section is a regular hexagon of side 4 cm. Calculate the total surface area of the prism. Answer(a) … cm2 [4] (b) Water fl ows through a cylindrical pipe of radius 0.74 cm. It fi lls a 12 litre bucket in 4 minutes. (i) Calculate the speed of the water through the pipe in centimetres per minute. Answer(b)(i) … cm/min [4]

15 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (ii) When the 12 litre bucket is emptied into a circular pool, the water level rises by 5 millimetres. Calculate the radius of the pool correct to the nearest centimetre. Answer(b)(ii) … cm [5] _____________________________________________________________________________________

16 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 10 (a) Write as a single fraction (i) 4 5 – x 5 2 , Answer(a)(i) … [2] (ii) 3 x 4 + + x 3 2 1 - . Answer(a)(ii) … [3] (b) Solve the simultaneous equations. 9x – 2y = 12 3x + 4y = –10 Answer(b) x = … y = … [3]

17 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (c) Simplify 2 2 9 9 7 21 x x x + + + . Answer(c) … [4] _____________________________________________________________________________________

18 0581/42/M/J/13 © UCLES 2013 For Examiner′s Use 11 Sidney draws the triangle OP1 P2. OP1 = 3 cm and P1 P2 = 1 cm. Angle OP1 P2 = 90°. (a) Show that OP2 = 10 cm. Answer(a) [1] (b) Sidney now draws the lines P2 P3 and OP3. Triangle OP2 P3 is mathematically similar to triangle OP1 P2. (i) Write down the length of P2 P3 in the form b a where a and b are integers. Answer(b)(i) P2 P3 = … cm [1] (ii) Calculate the length of OP3 giving your answer in the form d c where c and d are integers. Answer(b)(ii) OP3 = … cm [2] (c) Sidney continues to add mathematically similar triangles to his drawing. Find the length of OP5. Answer(c) OP5 = … cm [2] O P1 P2 1 cm 3 cm NOT TO SCALE O P1 P2 P3 1 cm 3 cm NOT TO SCALE O P1 P2 P3 P4 P5 1 cm 3 cm NOT TO SCALE

19 0581/42/M/J/13 © UCLES 2013 [Turn over For Examiner′s Use (d) (i) Show that angle P1OP2 = 18.4°, correct to 1 decimal place. Answer(d)(i) [2] (ii) Write down the size of angle P2OP3. Answer(d)(ii) Angle P2OP3 = … [1] (iii) The last triangle Sidney can draw without covering his fi rst triangle is triangle OP(n–1) Pn. O P1 Pn P2 P3 P4 P5 NOT TO SCALE P(n–1) Calculate the value of n. Answer(d)(iii) n = … [3] _____________________________________________________________________________________

20 0581/42/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 0581/42 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 42 © 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 Answers Mark Part Marks 1 (a) (i) 560 3 6 5 6 × + + [= 240] 2 Accept ‘of’ used instead of × M1 for 560 ÷ (5 + 6 + 3) (ii) 120 1 (b) 90 2 M1 for 240 8 3 × oe (c) (i) 96120 final answer 2 M1 for their(a)(ii) × 75 + (560 − their (a)(ii)) × 198 oe (ii) 187.5[0] final answer 3 M2 for 056 .0 1 198 + oe or M1 for (100 + 5.6)[%] = 198 oe seen (d) 184[.2…] 3 M2 for 100 5.9 5.9 75 .0 36 × − × oe or M1 for 100 5.9 75 .0 36 × × or 5.9 75 .0 36 − × [17.5] used implied by answer 84.2 or SC1 for final answer 284[.2..] (e) 69.4 and 69[.0] cao 3 SC2 for one correct or both correct but reversed M1 for two of 10.85, 10.95, 23.65 or 23.75 seen or 2(23.7 + 10.9) + 4(0.05) or 2(23.7 + 10.9) – 4(0.05)

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 42 © Cambridge International Examinations 2013 2 (a) (i) Translation,      − 8 5 oe 1,1 Brackets needed for vector Not (–5, 8), (–5 8) (ii) correct trapezium at (2, 2) (4 , 3) (4 , 5) (2, 5) 2 SC1 for reflection in x = −1 or vertices only (iii) correct trapezium at (4, 2) (5, 4) (7, 4) (7, 2) 3 M2 for 4 correct vertices on grid or in working or M1 for       − − − −       − 5 7 7 4 4 4 2 2 0 1 1 0 or SC1 for 3 vertices correct or complete shape in correct orientation but wrong position (b) (i) Shear x –axis (oe) invariant 2 1 1 1 (ii) rectangle at (−3, 2) (1, 2) (1, 8) (−3, 8) 2 SC1 for all vertices only or correct orientation and size, wrong position 3 (a) 0, 2, 0, − 3 3 B2 for 3 correct or B1 for 2 correct (b) Correct curve B4 B3FT for 8 points B2FT for 7 or 6 points B1FT for 5 or 4 points (c) y = –1 indicated x = 1.3 to 1.4 and 4.1 to 4.2 B1 B1 e.g. Could be mark[s] on curve isw other lines if not clearly used (d) (i) line drawn from (0, 2) to touch curve (2.5 to 2.75, 3 to 3.4) M1 A1 No daylight at point of contact If short, must cross at (0, 2) within ½ small square when extended (ii) rise/run e.g. (their y − 2)/their x 0.4 to 0.48 M1 A1 dep on attempt at a tangent from (0, 2) in (d)(i) and uses scales correctly Can be implied from answer– check on tangent for their rise for a run of 1 (½ small square) ww2 dep on attempt at a tangent from (0, 2) in (d)(i)

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 42 © Cambridge International Examinations 2013 4 (a) 227 or 226.95 to 227.01 2 M1 for 2 5.8 × π (b) 5.35 1 (c) 39.0[0] to 39.0[1] 2 M1 for sin [MOB]= 5.8 b their oe Dep on their b < 8.5 (d) 30.2 or 30.3 or 30.24 to 30.27 3 M2 for 5.8 2 360 39 4 360 × × × × − π oe or M1 for 5.8 2 360 × × × π a oe where 0 < a < 360 Implied by 5.78 to 5.79 or 11.5 to 11.6 or 23.14 to 23.15 or 23.1 or 23.2 or 41.83 to 41.84 or 41.8 (e) AB = BC TA = TC TB = TB 1 1 1 isw comments or reasons If 0 scored SC1 for “all three sides the same” oe [SSS] and no mention of angles 5 (a) x 27 final answer 1 (b) 2 25 − x final answer 1 (c) x x 27 4 2 25 = − − oe ( ) ( ) 2 27 2 4 25 − = − − x x x x oe ] 0 [ 54 8 25 27 4 2 = − − − + x x x x oe 0 27 3 2 2 = − −x x without error seen M1 M1 M1dep A1 FT their (b) − 4 = their (a) oe must be eqn in x FT x x 27 4 2 25 = + − oe only for 2nd and 3rd M mark If all on one side then condone omission of ‘= 0’ Dep on 2nd M1 Must see brackets expanded before this award and terms on one side of eqn Must see 0 54 6 4 2 = − − x x first (d) − 3, 4.5 3 B2 for ( )( ) 3 9 2 + − x x or SC1 for ( )( ) b x a x + + 2 where a and b are integers and a + 2b= − 3 or ab = − 27 (e) 6 cao 1

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 42 © Cambridge International Examinations 2013 6 (a) (i) 21 12 2 15 21 12 2 2 2 × × − + 44.41 to 44.42 M2 A2 M1 for 15² = 12² + 21² - 2.12.21cosM A1 for [cos =] 0.714 or 0.7142 to 0.7143 or 504 360 oe (ii) 88.2 or 88.15 to 88.19 2 M1 for 0.5 × 12 × 21 × sin (44.4) oe (b) 7.74 or 7.736 to 7.737…. www 4 B1 for 55 soi M2 82 sin ) sin( 4.6 × R their oe or M1 for 82 sin ) sin( 4.6 PR theirR = oe 7 (a) (i)       21 15 1 (ii) not possible oe 1 (iii) (2) final answer 2 M1 for 30 – 28 (iv)       0 0 13 4 1 (v)       − − 0 1 9 5 2 B1 for one correct row or column (b)       − − 2 1 4 3 2 1 or better isw 2 B1 for       − − 2 1 4 3 k seen or implied or       d c b a 2 1 seen 8 (a) hat 8 5 , 8 3 scarf 3 2 3 1 6 1 6 5 1 1 1 1 mark per pair in correct place (b) (i) 48 15 oe     16 5 2FT FT their 6 5 8 3 × correctly evaluated M1 6 5 8 3 × FT from their tree (ii) 24 5 2FT FT their 3 1 8 5 × correctly evaluated M1 3 1 8 5 × FT from their tree

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 42 © Cambridge International Examinations 2013 (iii) 48 13 cao 2 M1 for their 6 1 8 3 × + their (b)(ii) soi (c) 240 170 or 24 17 48 34 120 85 or or cao 3 M2 for 10 7 3 2 8 5 1 × × − FT their tree or 10 3 3 2 8 5 3 1 8 5 8 3 × × + × + oe or M1 for [“wears all” = ] 10 7 3 2 8 5 × × FT their tree seen 9 (a) 371 or 371.1... 4 M3 for ) 60 sin 4 4 5.0 6 2 ( ) 12 4 6 ( × × × × × + × × oe or M2 for area of 1 or 2 hexagons or M1 for area of one relevant triangle or trapezium or rectangle within hexagon If 0 scored SC1 for 288 shown (b) (i) 1740 or 1743.6 to 1744.2 4 M3 for ) 74 .0 ( 4 12000 2 × ÷ π oe or SC2 for figs 174[3..] or 174[4..] or B1 for 2 74 .0 × π seen [1.72..] or B1 for 12000 / 4 soi by 3000 (ii) 87 cao www 5 5 B4 for 87.39 to 87.43 or M3 for [r=] 5 12 figs π figs × oe or M2 for [ 2 r =] = 5 12 figs π figs oe or M1 for figs 5 12 2 figs r π × = 10 (a) (i) final answer 20 8 25 x − 2 M1 for 4 5 2 4 5 5 × × − × x or better seen (ii) final answer ( ) 3 3 9 5 2 2 + + + x x x 3 B1 for 3 6 2 2 − − + x x x soi and B1 for denom ( ) 3 3 + x or 9 3 + x seen (b) x = 3 2 oe or 0.667 or 0.6666 to 0.6667 y = −3 3 M1 for correct method to eliminate one variable A1 for x = 3 2 oe or 0.667 or 0.6666 to 0.6667 or y = −3

Page 7 Mark Scheme Syllabus Paper IGCSE – May/June 2013 0581 42 © Cambridge International Examinations 2013 (c) final answer 3 2 7 + x www 4 B1 for ( ) 3 7 + x in numerator and B2 for( )( ) 3 3 2 + + x x in denominator or SC1 for ( )( ) b x a x + + 2 where a and b are integers and a + 2b= 9 or ab = 9 After B1 scored, SC1 for final answer ( ) 5.1 5.3 5.1 2 7 + + x or x 11 (a) 32 + 12 1 Ignore attempt to evaluate 10 (b) (i) 3 10 final answer 1 (ii) 3 10 final answer 2 M1 for their 10 3 10 × or ( ) 2 2 10 3 10 +       their implied by 3.33 seen (c) 27 100 or 27 19 3 isw conversion or 3.7[03] to 3.7[04] 2 M1 for 3 n       × 3 10 oe where n is 3 or 4 or for [ ] 81 1000 4 = OP or for their (b)(ii) n       × 3 10 where n is 1 or 2 (d) (i) 18.43... 2 M1 for tan [P1OP2] = 3 1 oe (ii) 18.4[3…] 1 (iii) 20 3 SC2 for 19 or M1 for [ ] ... 3 4. 18 360