Cambridge IGCSE Mathematics (with coursework) 0581 — 2013 Oct/Nov 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) October/November 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 16 printed pages. [Turn over IB13 11_0581_42/FP © UCLES 2013 *0812510770* www.XtremePapers.com

2 0581/42/O/N/13 © UCLES 2013 For Examiner′s Use 1 Last year Mukthar earned $18 900 . He did not pay tax on $5500 of his earnings. He paid 24% tax on his remaining earnings. (a) (i) Calculate how much tax Mukthar paid last year. Answer(a)(i) $ … [2] (ii) Calculate how much Mukthar earned each month after tax had been paid. Answer(a)(ii) $ … [2] (b) This year Mukthar now earns $19 750.50 . Calculate the percentage increase from $18 900. Answer(b) … % [2] (c) Mukthar has $1500 to invest in one of the following ways. ● Account A paying simple interest at a rate of 4.1% per year ● Account B paying compound interest at a rate of 3.3% per year Which account will be worth more after 3 years and by how much? Answer(c) Account … by $ … [5] _____________________________________________________________________________________

3 0581/42/O/N/13 © UCLES 2013 [Turn over For Examiner′s Use 2 A D C B 2.4 m 8.6 m 1.8 m 6.46 m NOT TO SCALE The diagram shows the cross section, ABCD, of a ramp. (a) Calculate angle DBC. Answer(a) Angle DBC = … [2] (b) (i) Show that BD is exactly 3 m. Answer(b)(i) [2] (ii) Use the cosine rule to calculate angle ABD. Answer(b)(ii) Angle ABD = … [4] (c) The ramp is a prism of width 4 m. Calculate the volume of this prism. Answer(c) … m3 [3] _____________________________________________________________________________________

4 0581/42/O/N/13 © UCLES 2013 For Examiner′s Use 3 (a) Write as a single fraction in its simplest form. 3 1 x x 2 2 1 5 + - - Answer(a) … [3] (b) Expand and simplify. (2x – 3)2 – 3x(x – 4) Answer(b) … [4] (c) (i) Factorise. 2x2 + 5x – 3 Answer(c)(i) … [2] (ii) Simplify. 2 2 2 18 2 5 3 x x x - + - Answer(c)(ii) … [3] _____________________________________________________________________________________

5 0581/42/O/N/13 © UCLES 2013 [Turn over For Examiner′s Use 4 B A O 42° 8 cm 8 cm h cm NOT TO SCALE A wedge of cheese in the shape of a prism is cut from a cylinder of cheese of height h cm. The radius of the cylinder, OA, is 8 cm and the angle AOB = 42°. (a) (i) The volume of the wedge of cheese is 90 cm3. Show that the value of h is 3.84 cm correct to 2 decimal places. Answer(a)(i) [4] (ii) Calculate the total surface area of the wedge of cheese. Answer(a)(ii) … cm2 [5] (b) A mathematically similar wedge of cheese has a volume of 22.5 cm3. Calculate the height of this wedge. Answer(b) … cm [3] _____________________________________________________________________________________

6 0581/42/O/N/13 © UCLES 2013 For Examiner′s Use 5 (a) Complete the table of values for y = 2 x 2 1 3 - - x x . x –3 –2 –1 –0.5 –0.3 0.3 0.5 1 2 3 y 9.6 6 26.5 18.0 –2 –6 –9.1 [3] (b) Draw the graph of y = 2 x 2 1 3 - - x x for –3 Y x Y –0.3 and 0.3 Y x Y 3 . y x 30 25 20 15 10 5 –5 –10 0 –1 –2 –3 3 2 1 [5]

7 0581/42/O/N/13 © UCLES 2013 [Turn over For Examiner′s Use (c) Use your graph to solve these equations. (i) 2 x 2 1 3 - - x x = 0 Answer(c)(i) x = … [1] (ii) 2 3 . x 2 1 7 5 - - - x x = 0 Answer(c)(ii) x = … or x = … or x = … [3] (d) (i) By drawing a suitable straight line on the graph, solve the equation 2 3 10 3 x x 2 1 - - = - x x . Answer(d)(i) x = … or x = … [4] (ii) The equation 2 3 10 3 x x 2 1 - - = - x x can be written in the form ax2 + bx + c = 0 where a, b and c are integers. Find the values of a, b and c. Answer(d)(ii) a = … , b = … , c = … [3] _____________________________________________________________________________________

8 0581/42/O/N/13 © UCLES 2013 For Examiner′s Use 6 E N L A R G E M E N T Prettie picks a card at random from the 11 cards above and does not replace it. She then picks a second card at random and does not replace it. (a) Find the probability that she picks (i) the letter L and then the letter G, Answer(a)(i) … [2] (ii) the letter E twice, Answer(a)(ii) … [2] (iii) two letters that are the same. Answer(a)(iii) … [2]

9 0581/42/O/N/13 © UCLES 2013 [Turn over For Examiner′s Use (b) Prettie now picks a third card at random. Find the probability that the three letters (i) are all the same, Answer(b)(i) … [2] (ii) do not include a letter E, Answer(b)(ii) … [2] (iii) include exactly two letters that are the same. Answer(b)(iii) … [5] _____________________________________________________________________________________

10 0581/42/O/N/13 © UCLES 2013 For Examiner′s Use 7 Noma fl ies from Johannesburg to Hong Kong. Her plane leaves Johannesburg at 18 45 and arrives in Hong Kong 13 hours and 25 minutes later. The local time in Hong Kong is 6 hours ahead of the time in Johannesburg. (a) At what time does Noma arrive in Hong Kong? Answer(a) … [2] (b) Noma sleeps for part of the journey. The time that she spends sleeping is given by the ratio sleeping : awake = 3 : 4 . Calculate how long Noma sleeps during the journey. Give your answer in hours and minutes. Answer(b) … h … min [2]

11 0581/42/O/N/13 © UCLES 2013 [Turn over For Examiner′s Use (c) (i) The distance from Hong Kong to Johannesburg is 10 712 km. The time taken for the journey is 13 hours and 25 minutes. Calculate the average speed of the plane for this journey. Answer(c)(i) … km/h [2] (ii) The plane uses fuel at the rate of 1 litre for every 59 metres travelled. Calculate the number of litres of fuel used for the journey from Johannesburg to Hong Kong. Give your answer in standard form. Answer(c)(ii) … litres [4] (d) The cost of Noma’s journey is 10 148 South African Rand (R). This is an increase of 18% on the cost of the journey one year ago. Calculate the cost of the same journey one year ago. Answer(d) R … [3] _____________________________________________________________________________________

12 0581/42/O/N/13 © UCLES 2013 For Examiner′s Use 8 f(x) = 4x + 3 g(x) = 1 x 7 + (x ¸ –1) h(x) = x2 + 5x (a) Work out (i) h(–3), Answer(a)(i) … [1] (ii) hg(13). Answer(a)(ii) … [2] (b) Find f –1(x). Answer(b) f –1(x) = … [2]

13 0581/42/O/N/13 © UCLES 2013 [Turn over For Examiner′s Use (c) (i) Solve the equation f(x) = 23. Answer(c)(i) x = … [2] (ii) Solve the equation h(x) = 7. Show all your working and give your answers correct to 2 decimal places. Answer(c)(ii) x = … or x = … [5] _____________________________________________________________________________________

14 0581/42/O/N/13 © UCLES 2013 For Examiner′s Use 9 A C D B y x 8 7 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 –8 (a) Describe fully the single transformation that maps triangle A onto (i) triangle B, Answer(a)(i) … [2] (ii) triangle C, Answer(a)(ii) … [2] (iii) triangle D. Answer(a)(iii) … [3]

15 0581/42/O/N/13 © UCLES 2013 [Turn over For Examiner′s Use (b) On the grid, draw (i) the rotation of triangle A about (6, 0) through 90° clockwise, [2] (ii) the enlargement of triangle A by scale factor –2 with centre (0, –1), [2] (iii) the shear of triangle A by shear factor –2 with the y-axis invariant. [2] (c) Find the matrix that represents the transformation in part (b)(iii). Answer(c) f p [2] _____________________________________________________________________________________ Question 10 is printed on the next page.

16 0581/42/O/N/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. © UCLES 2013 For Examiner′s Use 10 Complete the table for the following sequences. The fi rst row has been completed for you. Sequence Next two terms nth term 1 5 9 13 17 21 4n – 3 (a) 12 21 30 39 [3] (b) 80 74 68 62 [3] (c) 1 8 27 64 [2] (d) 2 10 30 68 [2]

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 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 October/November 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 – October/November 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 Correct answer Mark Part marks 1 (a) (i) 3216 Final answer 2 M1 for (18900 – 5500) × 0.24 oe (ii) 1307 Final answer 2FT FT (18900 – their (a)(i)) ÷ 12 correctly evaluated M1 for (18900 – their (a)(i)) ÷ 12 (b) 4.5[%] nfww 2 M1 for 100 18900 ] 18900 [ 50 . 19750 × − or 18900 18900 50 . 19750 − (c) A by 31.05… or 31.04 to 31.05 or 31.[0] 31.1[0] 5 M1 for 1500 × 4.1/100 × 3 [+ 1500] oe M1 for 1500 × 1.0333 [– 1500] oe A1 for 1684.5 or 184.5 or 1653[.45..] or 153[.45..] and M1dep for subtraction of their amounts or their interests 2 (a) 36.9° or 36.86 to 36.87 2 M1 for tan[DBC] = 1.8/2.4 oe (b) (i) 1.8² + 2.4² leading to 9 2 M1 for 1.8² + 2.4² or better (ii) [cosABD) =] 3 46 .6 2 6.8 3 46 .6 2 2 2 × × − + 127 or 126.8… M2 A2 M1 for correct cos rule but implicit version A1 for –0.599… After 0 scored, SC2 nfww for answer 127 or 126.8 to 126.96 from other methods or no working shown (c) 39.6 or 39.7 or 39.59 to 39.68 3 M2 for ½ (2.4 + 8.6) × 1.8 × 4 oe Or M1 for ) 6.8 4.2 ( 2 8.1 + oe soi by 9.9 to 9.92

Page 3 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0581 42 © Cambridge International Examinations 2013 3 (a) 10 7 4 − x final answer nfww 3 M2 for 5 2 )1 3 ( 2 )1 2 ( 5 × + − − x x or 2 5 )1 2 ( 5 × − x – 2 5 )1 3 ( 2 × + x or M1 for attempt to convert to common denominator of 10 or multiple of 10 with one error in numerator (b) x² + 9 final answer nfww 4 B3 for 4x² – 6x – 6x + 9 – 3x² +12x or correct answer given and then spoilt or B1 for 4x² – 6x – 6x + 9 seen and B1 for – 3x² +12x or – (3x² – 12x) seen (c) (i) (2x – 1)(x + 3) isw solving 2 M1 for (2x + a)(x + b) where ab = –3 or 2b + a = 5 with integers a and b (ii) )3 ( 2 1 2 − − x x or 6 2 1 2 − − x x final answer nfww 3 M2 for 2(x + 3)(x – 3) or (2x – 6)(x + 3) or (2x + 6)(x – 3) seen or M1 for 2(x² – 9) seen 4 (a) (i) 90 ÷ (42/360 × π × 82) o.e. 3.836 to 3.837 M3 A1 M2 for 42/360 × π × 82 × h = 90 or M1 for 42/360 × π × 82 (ii) 131 or 130.75 to 130.9 nfww 5 M2 for 42/360 × π × 2 × 8 × 3.84 oe [22.48 to 22.53] or M1 for 42/360 × π × 2 × 8 oe soi [5.86 to 5.87] and M1 for 2 × (8 × 3.84) [61.37 to 61.44] and M1 for 2 × (42/360 × π × 82) [46.88 to 47] (b) 2.42 or 2.416 to 2.419 3 M2 for 3.84 × 3 90 5. 22 oe or h = 3 3 90 5. 22 84 .3 × or M1 for 3 90 5. 22 oe or 3 5. 22 90 oe seen or 3 3 84 .3 h = 5. 22 90 oe

Page 4 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0581 42 © Cambridge International Examinations 2013 5 (a) 7, 11.5, 4.5 1,1,1 (b) Correct curve cao 5 B3FT for 10 correct plots, on correct vertical grid line and within correct 2 mm square vertically Or B2FT for 8 or 9 correct plots Or B1FT for 6 or 7 correct plots and B1 indep for two separate branches on either side of y-axis (c) (i) 0.69 < x < 0.81 1 (ii) –2.3 < x < –2.2 –0.8 < x < –0.6 0.35 < x < 0.5 3 B1 for each correct After 0 scored, allow SC1 for drawing line y = 7.5 long enough to cross curve at least once (d) (i) y = 10 – 3x ruled correctly –0.55 < x < –0.45 0.35 < x < 0.45 B2 B1dep B1dep long enough to cross curve twice. B1 for ruled line gradient –3 or y intercept at 10 but not y = 10 Or B1 for ‘correct’ but freehand Dependent on at least B1 scored for line After 0 scored, SC2 for –0.5 and 0.4 [from solving equation] (ii) 10 1 –2 or –10 –1 2 3 B2 for 2 – x – 10x2 [= 0] oe Or B1 for 0 10 1 2 2 = − −x x oe Correctly eliminating – 3x Or B1 for 2 – x – 3x3 = 10x2 – 3x3 oe Correctly clearing fractions

Page 5 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0581 42 © Cambridge International Examinations 2013 6 (a) (i) 110 1 oe 2 M1 for 10 1 11 1 × (ii) 110 6 oe     55 3 2 M1 for 10 2 11 3 × (iii) 110 8 oe     55 4 2FT FT their (a)(ii) + 10 1 11 2 × correctly evaluated or M1 their (a)(ii) + 10 1 11 2 × (b) (i) 990 6 oe     165 1 2 M1 for 9 1 10 2 11 3 × × (ii) 990 336 oe     165 56 2 M1 for 9 6 10 7 11 8 × × (iii) 990 198 oe     5 1 5 M4 for          × × +       × × 9 9 10 1 11 2 3 9 8 10 2 11 3 3 oe or M3 for          × ×       × × 9 9 10 1 11 2 3 9 8 10 2 11 3 3 or oe Or M1 for 9 8 10 2 11 3 × × oe seen and M1 for    × × 9 9 10 1 11 2 oe seen

Page 6 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0581 42 © Cambridge International Examinations 2013 7 (a) 14 10 or 2 10 pm final answer 2 M1 for (0)8 10 oe or answer 14 hours and 10 minutes or answer 2 10 [am] (b) 5 hours 45 minutes cao 2 M1 for 345 [mins] seen or for 805 /7 × 3 oe or 5.75 seen (c) (i) 798 or 798.2 to 798.4…. 2 M1 for 10712 / 60 25 13 or 10712 ÷ 13.4… (ii) 1.82 × 105 or 1.815 × 105 to 1.816 × 105 4 B3 for 182000 or 181500 to 181600 seen or M2 for 10712000/59 oe or M1 for figs 10712/figs 59 soi by figs 182 or figs 1815 to 1816 and B1 FT for their number of litres correctly converted to standard form rounded to 3sf or better (d) 8600 3 M2 for 10148 ÷ 1.18 oe or M1 for 10148 associated with 118[%] 8 (a) (i) –6 1 (ii) 2.75 oe 2 M1 for [g(x) =] 0.5 or 7/14 Or       + +       + 1 7 5 1 7 2 x x oe (b) 4 3 − x or 4 x – 4 3 Final answer 2 M1 for y – 3 = 4x or better or x = 4y + 3 or better or 4 y = 4 3 + x or flowchart with – 3 then ÷ 4 (c) (i) 5 2 M1 for 4x = 23 – 3 or x + 4 3 = 4 23 or better (ii) x² + 5x – 7 = 0 )1( 2 ) 7 )( 1( 4 5 5 2 − − ± − oe 1.14 and –6.14 final answers B1 B1 B1 B1 B1 May be implied by correct values in formula B1 for ) 7 )( 1( 4 52 − − or better [53] If in form r q p + or r q p − , B1 for –5 and 2(1) or better No recovery of full line unless seen Or SC1 for 1.1 or 1.140…. and –6.1 or – 6.140 … Or answers –1.14 and 6.14

Page 7 Mark Scheme Syllabus Paper IGCSE – October/November 2013 0581 42 © Cambridge International Examinations 2013 9 (a) (i) Reflection x = –2 oe 2 B1 for either (ii) Translation      − 2 7 oe 2 B1 for either (iii) Stretch x-axis oe invariant [factor] 3 3 B1 for each (b) (i) Triangle with coords at (8, 2) (7, 3) and (7, 5) 2 B1 for rotation about (6, 0) but 90° anticlockwise Or for rotation 90° clockwise around any point (ii) Triangle with coords at (–2, –5) (–6, –5) and (–8, –7) 2 B1 for 2 correct points or for enlargement of SF –2 any centre (iii) Triangle with coords at (1, –1) (4, –6) and (3, –5) 2 B1 for 2 correct points or coordinates of 2 points shown (c)       − 1 2 0 1 2 B1 for one row or one column correct but not identity matrix. Or SC1 for       − 1 0 2 1 10 (a) 48 and 57, 9n + 3 oe (b) 56 and 50, 86 – 6n oe (c) 125 and 216, n3 oe (d) 130 and 222 n3 + n oe 1 2 1 2 1 1 1 1FT B1 for 9n + k oe B1 for k – 6n oe FT their (c) + n dep on expression in n in (c)