Cambridge IGCSE Mathematics (with coursework) 0581 — 2012 May/June Paper 4 · Variant 2

This document consists of 16 printed pages. IB12 06_0581_42/FP © UCLES 2012 [Turn over *4928945136* UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0581/42 Paper 4 (Extended) May/June 2012 2 hours 30 minutes Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments Mathematical tables (optional) Tracing paper (optional) 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 fluid. 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 specified in the question, and if the answer is not exact, give the answer to three significant figures. 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. www.XtremePapers.com

2 © UCLES 2012 0581/42/M/J/12 For Examiner's Use 1 Mathematics mark 30 50 35 25 5 39 48 40 10 15 English mark 26 39 35 28 9 37 45 33 16 12 The table shows the test marks in Mathematics and English for 10 students. (a) (i) On the grid, complete the scatter diagram to show the Mathematics and English marks for the 10 students. The first four points have been plotted for you. 50 40 30 20 10 0 5 10 15 20 25 30 35 40 45 50 English mark Mathematics mark [2] (ii) What type of correlation does your scatter diagram show? Answer(a)(ii) [1] (iii) Draw a line of best fit on the grid. [1] (iv) Ann missed the English test but scored 22 marks in the Mathematics test. Use your line of best fit to estimate a possible English mark for Ann. Answer(a)(iv) [1] (b) Show that the mean English mark for the 10 students is 28. Answer(b) [2] (c) Two new students do the English test. They both score the same mark. The mean English mark for the 12 students is 31. Calculate the English mark for the new students. Answer(c) [3]

3 © UCLES 2012 0581/42/M/J/12 [Turn over For Examiner's Use 2 (a) In a sale, Jen buys a laptop for $351.55. This price is 21% less than the price before the sale. Calculate the price before the sale. Answer(a) $ [3] (b) Alex invests $4000 at a rate of 8% per year simple interest for 2 years. Bob invests $4000 at a rate of 7.5% per year compound interest for 2 years. Who receives more interest and by how much? Answer(b) receives $ more interest. [6]

4 © UCLES 2012 0581/42/M/J/12 For Examiner's Use 3 Pablo plants x lemon trees and y orange trees. (a) (i) He plants at least 4 lemon trees. Write down an inequality in x to show this information. Answer(a)(i) [1] (ii) Pablo plants at least 9 orange trees. Write down an inequality in y to show this information. Answer(a)(ii) [1] (iii) The greatest possible number of trees he can plant is 20. Write down an inequality in x and y to show this information. Answer(a)(iii) [1] (b) Lemon trees cost $5 each and orange trees cost $10 each. The maximum Pablo can spend is $170. Write down an inequality in x and y and show that it simplifies to x + 2y Y 34. Answer (b) [1] (c) (i) On the grid opposite, draw four lines to show the four inequalities and shade the unwanted region.

5 © UCLES 2012 0581/42/M/J/12 [Turn over For Examiner's Use y x 24 22 20 18 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 [7] (ii) Calculate the smallest cost when Pablo buys a total of 20 trees. Answer(c)(ii) $ [2]

6 © UCLES 2012 0581/42/M/J/12 For Examiner's Use 4 (a) B A F O C D E 42° NOT TO SCALE A, B, C, D, E and F are points on the circumference of a circle centre O . AE is a diameter of the circle. BC is parallel to AE and angle CAE = 42°. Giving a reason for each answer, find (i) angle BCA, Answer(a)(i) Angle BCA = Reason [2] (ii) angle ACE, Answer(a)(ii) Angle ACE = Reason [2] (iii) angle CFE, Answer(a)(iii) Angle CFE = Reason [2] (iv) angle CDE. Answer(a)(iv) Angle CDE = Reason [2]

7 © UCLES 2012 0581/42/M/J/12 [Turn over For Examiner's Use (b) O P Q 12 cm 5 cm NOT TO SCALE In the diagram, O is the centre of the circle and PQ is a tangent to the circle at P. OP = 5 cm and OQ = 12 cm. Calculate PQ. Answer(b) PQ = cm [3] (c) D C B A G F E NOT TO SCALE In the diagram, ABCD and DEFG are squares. (i) In the triangles CDG and ADE, explain with a reason which sides and/or angles are equal. Answer (c)(i) [3] (ii) Complete the following statement. Triangle CDG is to triangle ADE. [1]

8 © UCLES 2012 0581/42/M/J/12 For Examiner's Use 5 (a) In Portugal, Miguel buys a book about planets. The book costs €34.95. In England the same book costs £27.50. The exchange rate is £1 = €1.17. Calculate the difference in pounds (£) between the cost of the book in Portugal and England. Answer(a) £ [2] (b) In the book, the distance between two planets is given as 4.07 × 1012 kilometres. The speed of light is 1.1 × 109 kilometres per hour. Calculate the time taken for light to travel from one of these planets to the other. Give your answer in days and hours. Answer(b) days hours [3] (c) In one of the pictures in the book, a rectangle is drawn. The rectangle has length 9.3 cm and width 5.6 cm, both correct to one decimal place. (i) What is the lower bound for the length? Answer(c)(i) cm [1] (ii) Work out the lower and upper bounds for the area of the rectangle. Answer(c)(ii) Lower bound = cm2 Upper bound = cm2 [2]

9 © UCLES 2012 0581/42/M/J/12 [Turn over For Examiner's Use 6 (a) x° 2x° 114° (x – 10)° NOT TO SCALE Find the value of x. Answer(a) x = [3] (b) (i) Write the four missing terms in the table for sequences A, B, C and D. Term 1 2 3 4 5 n Sequence A – 4 2 5 8 3n – 7 Sequence B 1 4 9 16 25 Sequence C 5 10 15 20 25 Sequence D 6 14 24 36 50 [4] (ii) Which term in sequence D is equal to 500? Answer(b)(ii) [2] (c) Simplify 4 7 2 16 2 2 − + − x x x . Answer(c) [4]

10 © UCLES 2012 0581/42/M/J/12 For Examiner's Use 7 (a) P is the point (2, 5) and =       −2 3 . Write down the co-ordinates of Q. Answer(a) ( , ) [1] (b) D C E M B A O c 3a NOT TO SCALE O is the origin and OABC is a parallelogram. M is the midpoint of AB. = c, = 3a and CE = 3 1 CB. OED is a straight line with OE : ED = 2 : 1 . Find in terms of a and c, in their simplest forms (i) , Answer(b)(i) = [1] (ii) the position vector of M, Answer(b)(ii) [2] (iii) , Answer(b)(iii) = [1] (iv) . Answer(b)(iv) = [2] (c) Write down two facts about the lines CD and OB. Answer (c) [2]

11 © UCLES 2012 0581/42/M/J/12 [Turn over For Examiner's Use 8 In all parts of this question give your answer as a fraction in its lowest terms. (a) (i) The probability that it will rain today is 3 1 . What is the probability that it will not rain today? Answer(a)(i) [1] (ii) If it rains today, the probability that it will rain tomorrow is 5 2 . If it does not rain today, the probability that it will rain tomorrow is 6 1 . Complete the tree diagram. Today Tomorrow Rain No rain Rain No rain No rain Rain [2] (b) Find the probability that it will rain on at least one of these two days. Answer(b) [3] (c) Find the probability that it will rain on only one of these two days. Answer(c) [3]

12 © UCLES 2012 0581/42/M/J/12 For Examiner's Use 9 E H F G Scale 1 : 10 000 The diagram is a scale drawing of a park EFGH. The scale is 1 : 10 000. A statue is to be placed in the park so that it is • nearer to G than to H • nearer to HG than to FG • more than 550 metres from F. Construct accurately the boundaries of the region R in which the statue can be placed. Leave in all your construction arcs and shade the region R. [7]

13 © UCLES 2012 0581/42/M/J/12 [Turn over For Examiner's Use 10 (a) Simplify (i) (2x2y3)3, Answer(a)(i) [2] (ii) 3 1 _ 6 27       x . Answer(a)(ii) [3] (b) Multiply out and simplify. (3x – 2y)(2x + 5y) Answer(b) [3] (c) Make h the subject of (i) V = πr3 + 2πr2h, Answer(c)(i) h = [2] (ii) V = h 3 . Answer(c)(ii) h = [2] (d) Write as a single fraction in its simplest form. 2 x + 3 5x – 4 7x Answer(d) [2]

14 © UCLES 2012 0581/42/M/J/12 For Examiner's Use 11 (a) Calculate the area of a circle with radius 12 cm. Answer(a) cm2 [2] (b) 22° 12 cm 7 cm NOT TO SCALE A circular cake has radius 12 cm and height 7 cm. The uniform cross-section of a slice of the cake is a sector with angle 22°. The top and the curved surface of the slice, shaded in the diagram, are covered with chocolate. Calculate the area of the slice which is covered with chocolate. Answer(b) cm2 [5]

15 © UCLES 2012 0581/42/M/J/12 [Turn over For Examiner's Use (c) 50° 50° 100° A B C D 31 cm 22 cm NOT TO SCALE The frame of a child’s bicycle is made from metal rods. ABC is an isosceles triangle with base 22 cm and base angles 50°. Angle ACD = 100° and CD = 31 cm. Calculate the length AD. Answer(c) AD = cm [6] Question 12 is printed on the next page.

16 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. © UCLES 2012 0581/42/M/J/12 For Examiner's Use 12 (a) The cost of 1 kg of tomatoes is $x and the cost of 1 kg of onions is $y. Ian pays a total of $10.70 for 10 kg of tomatoes and 4 kg of onions. Jao pays a total of $10.10 for 8 kg of tomatoes and 6 kg of onions. Write down simultaneous equations and solve them to find x and y. Answer(a) x = y = [6] (b) Solve 2x2– 5x – 8 = 0 . Give your answers correct to 2 decimal places. Show all your working. Answer(b) x = or x = [4]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2012 question paper for the guidance of teachers 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 must be read in conjunction with the question papers and the report on the examination. • Cambridge will not enter into discussions or correspondence in connection with these mark schemes. Cambridge is publishing the mark schemes for the May/June 2012 question papers for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses. www.XtremePapers.com

Page 2 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 42 © University of Cambridge International Examinations 2012 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) 6 correct plots 2 P1 for 4 or 5 correct plots. (ii) Positive 1 (iii) Line of best fit 1 Ruled line at least from x = 5 to x = 48, with at least 3 points on each side and cuts axes between (5, 0) and (0, 20) (iv) English (integer) value on line at M = 22 1ft Strict ft from their single ruled line 5 ø x ø 48. (b) (26 + 39 + 35 + 28 + 9 + 37 + 45 + 33 + 16 + 12) ÷ 10 M2 M1 for 26 + 39 + 35 + 28 + 9 + 37 + 45 + 33 + 16 + 12, condone one slip or SC1, for at least 2 values eg (26 + 39 +…) ÷ 10 (c) 46 cao www 3 3 M2 for (31 × 12 – 28 × 10) ÷ 2 soi by 92 ÷ 2 or M1 for 31 × 12 soi by 372 or 92 2 (a) 445 final answer www 3 3 M2 for 351.55 ÷ (1 – 0.21) oe or M1 for 351.55 = (100 – 21) (%) (b) 640 or 4640 4622.5 or 622.5 Alex by 17.5(0) cao final answer www 6 2 2 2 M1 for 4000 × 0.08 × 2 oe M1 for 4000 × (1.075) 2 oe or 4000 × 0.075 (= 300) and (4000 + their 300) × 0.075 and total interest = the sum of their 2 interests. M1 for S I amount – C I amount or reverse or simple interest – compound interest or reverse

Page 3 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 42 © University of Cambridge International Examinations 2012 3 (a) (i) x > 4 1 (ii) y > 9 1 (iii) x + y < 20 1 (b) 5x + 10y < 170 seen 1 (c) (i) x = 4 ruled y = 9 ruled x + y = 20 ruled x + 2y = 34 ruled Correct region indicated cao 1 1 2 2 1 Each line long enough to enclose their region Condone good freehand or dotted y = 9 must be between 8.8 and 9.2 B1 for gradient = – 1 or y intercept = 20 or x intercept = 20. Exclude lines parallel to either axis. B1 for y intercept = 17 or x intercept = 34. Exclude lines parallel to either axis. Dependent on all 6 marks for the 4 lines. (ii) 145 cao (from 11, 9) www 2 2 M1 for using 5x + 10y when x + y = 20 and integers (x, y) is in their region

Page 4 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 42 © University of Cambridge International Examinations 2012 4 In all parts of (a) candidates may refer to angles marked in diagram. Allow if clear even if reason is more complicated as long as it is full. Reasons dependent on correct answers (a) (i) 42 Alternate oe 1 1 Not alternate segment (ii) 90 semicircle oe 1 1 Allow diameter (iii) 42 same segment oe 1 1 same arc (iv) 138 cyclic quad oe 1 1 key words must not be spoiled (b) 10.9 (10.90 to 10.91) www 3 3 M2 for 2 2 5 12 − oe i.e explicit or M1 for 12 2 = 5 2 + PQ 2 oe i.e implicit Allow full marks for 119 as final answer Use of trig method must be complete to explicit expression for possible M2 (c) (i) AD = CD and DE = DG (Angle) CDG = (angle)ADE (Sides of) square or 90° + angle ADG oe 1 1 R1 Extra pair of sides loses this mark. Extra pair of angles loses this mark As in (a), for all 3 marks allow references to diagram if completely clear. R mark dep on at least one pair of sides stated or pair of angles stated (ii) Congruent 1

Page 5 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 42 © University of Cambridge International Examinations 2012 5 (a) (£) 2.37 or 2.371 to 2.372 www 2 2 M1 for 34.95 ÷ 1.17 implied by 29.87…or 29.9 or SC1 for 2.77 or 2.78 or 2.775 (b) 154 days 4 hours cao 3 M1 for 4.07 × 1012 ÷ (1.1 × 109) implied by figs 37 or 154. (….) A1 for 3700 seen or 3.7 × 103 seen or 154 6 1 oe or 154 rem 4 (c) (i) 9.25 1 (ii) Lower = 51.3375 final answer Upper = 52.8275 final answer 1 1 After 0 scored SC1 for answers reversed or 9.35 and 5.65 seen or 51.3375 and 52.8275 seen 6 (a) (x =) 64 www 3 3 B2 for 10 114 360 2 + − = + + x x x or better or M1 for = − + + + 10 114 2 x x x 360 (b) (i) –1 n 2 oe 5n oe n 2 + 5n oe 1 1 1 1 (ii) 20 2 M1 for their n 2 + 5n = 500 or 20 and 25 seen (c) Final answer 1 2 4 − − x x cao www 4 4 B1 for (x – 4)(x + 4) B2 for (2x – 1)(x + 4) or SC1 for (2x + a)(x + b) where either a + 2b = 7 or ab = – 4 7 (a) (5, 3) 1 (b) (i) 3a + c 1 (ii) 3a + 2 1 c or 2 1 (6a + c ) 2 M1 for OM oe e.g OA+AM or correct unsimplified answer (iii) a + c 1 (iv) 2 3 a + 2 1 c or 2 1 (3a + c) 2 M1 for – c + 2 3× their (iii) or a + 2 1× their (iii) or correct unsimplified answer or any correct route e.g. CE + ED (c) (CD) parallel (to OB) oe cao CD = 2 1 OB oe cao 1dep 1dep Part (c) dependent on simplified (i) and (iv) Dep on (i) = k × (iv) Dep on (i) = 2 × (iv) must be scalars

Page 6 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 42 © University of Cambridge International Examinations 2012 8 Throughout question, penalise non-reduced fraction only once; isw any conversion and allow decimals in working and on branches but not final answers if fractions not seen. (a) (i) 3 2 1 (ii) 3 1 , 3 2 , 5 2 , 5 3 , 6 1 , 6 5 correctly placed 2 B1 for 3 1 and 3 2 and 5 3 or 6 5 correctly placed For method marks in (b) and (c), ft tree with each probability 0< p <1 (b) 9 4 cao www 3 3 M2 for 1 – 3 2 × 6 5 or 3 1 + 3 2 × 6 1 or 3 1 × 5 2 + 3 1 × 5 3 + 3 2 × 6 1 M1 for 3 1 + 3 2 × 6 5 or two of 3 1 × 5 2 , 3 1 × 5 3 , 3 2 × 6 1 added (c) 45 14 cao www 3 3 M2 for 3 1 × 5 3 + 3 2 × 6 1 or their 9 4 – 3 1 × 5 2 M1 for one of 3 1 × 5 3 or 3 2 × 6 1 from a maximum of two products added. 9 Accurate ruled perp. bisector with correct intersecting arcs Accurate ruled angle bisector with correct intersecting arcs Compass drawn arc centre F radius 5.5 cm long enough to enclose region Correct region indicated cao 2 2 2 1 B1 for accurate with no/wrong arcs or M1 for correct intersecting arcs Ignore one extra perp. bisector B1 for accurate with no/wrong arcs or M1 for correct intersecting arcs Ignore one extra angle bisector M1 for compass drawn arc centre F Accept dotty lines but not freehand for all three

Page 7 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 42 © University of Cambridge International Examinations 2012 10 (a) (i) 8x 6 y 9 final answer 2 B1 for any two of 8, x 6, y 9 in a single term in answer (ii) 3 2 x oe but not 2 3 1 − x oe final answer 3 B2 for 2 3 x or 2 3 − x or 2 3 1 − x as answer or B1 for 27 6 x oe as answer or 3 6 1 27 x seen or SC1 for 3 or x 2 or x – 2 seen in answer (b) 6x 2 + 11xy – 10y 2 final answer 3 B2 for 3 of 6x 2 – 4xy + 15xy – 10y 2 (11xy implies 2 terms) or B1 for 2 of 6x 2 – 4xy + 15xy – 10y 2 (c) (i) 2 3 2 r r V π π − or 2 2 2 r r V − π oe but not triple fractions final answer 2 M1 for correct subtraction or correct division by 2 2 r π seen (ii) 3 2 V final answer 2 B1 for V 2 = 3h or 3 V = h or 2 3       = V h (d) 12 5x final answer 2 B1 for 2 of 12 6x , 12 20x , 12 21x − oe implied by 24 10x ie 2 with common denominator = at least 6 11 (a) 452 or 452.1 to 452.4… 2 M1 for π × 12 2 Allow full marks for 144π as final answer (b) 59.9 or 59.86 to 59.91 cao www 5 5 M1 for π × 24 × 7 (soi by 527 to 528) oe or 24 360 22 × π × oe (soi by 4.60 to 4.61) and M1 dep for 360 22 × π × 24 × 7 (soi by 32.2 to 32.3) and M1 for × 360 22 their (a) oe may restart (soi by 27.6 to 27.7) and M1 dep on M3 for adding two areas (c) 11 ÷ cos50 soi by 17.(11…) oe (their AC) 2 + 312 – 2 × their AC × 31cos100 art 37.9 cao www 6 M2 M2 A2 M1 for cos 50 = AC 11 oe i.e. implicit M1 for implicit cos rule A1 for 1433 to 1443

Page 8 Mark Scheme: Teachers’ version Syllabus Paper IGCSE – May/June 2012 0581 42 © University of Cambridge International Examinations 2012 12 (a) 10x + 4y = 10.7 oe 8x + 6y = 10.1 oe Multiplying or dividing equation(s) by number(s) suitable for elimination Elimination of one variable x = 0.85 cao y = 0.55 cao 1 1 M1 M1 A1 A1 Allow one arithmetic error. If substitution, correctly making one variable the subject of one equation. Allow one arithmetic error. If substitution method then M is for the actual substitution. SC1 for correct fractions After M0, SC2 for both correct answers If working in cents, likely mark is 0 for equations, M2 for method, A2 if answers converted to dollars, A1 if left in cents (b) ( ) 2.2 8 .2.4 5 5 2 − − − ± − − B2 B1 for ( ) 8 .2.4 5 2 − − − ( 89 ) B1 for r p + or r p − with p = – – 5 or 5 and r = 2 × 2 or 4 Completing the square B1 for 2 4 5       − x and B1 for 16 25 4 + 3.61 or –1.11 final answer B1B1 After B0 B0 for answers, SC1 for 3.6 or 3.608… and – 1.1 or – 1.108 or 3.61 and –1.11 seen Correct answers without working score max 2