Cambridge IGCSE Mathematics (with coursework) 0581 — 2005 May/June Paper 3 · Variant 1

This document consists of 15 printed pages and 1 blank page. IB05 06_0580_03/5RP  UCLES 2005 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS Paper 3 (Core) 0580/03 0581/03 Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments May/June 2005 Mathematical tables (optional) Tracing paper (optional) 2 hours 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 in the spaces provided on the Question Paper. 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 THE BARCODE. DO NOT WRITE IN THE GREY AREAS BETWEEN THE PAGES. Answer all questions. If working is needed for any question it must be shown below that question. The number of marks is given in brackets [ ] at the end of each question or part question. The total number of marks for this paper is 104. 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. Given answers in degrees to one decimal place. For π , use either your calculator value or 3.142. Candidate Name Centre Number Candidate Number For Examiner's Use *058003* www.XtremePapers.com

2 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use 1 Juana is travelling by plane from Spain to England. (a) Her case weighs 17.2 kilograms. The maximum weight allowed is 20 kilograms. By how much is the weight of her case below the maximum allowed? Answer (a) kg [1] (b) She changes 150 euros (€) into pounds (£). The exchange rate is €1 = £0.71. Calculate how much she receives. Answer (b) £ [1] (c) She travels from her home to the airport by train. She catches a train at 09 55 and the journey takes 45 minutes. (i) Write down the time she arrives at the airport. Answer (c)(i) [1] (ii) She has to wait until 12 10 to get on her plane. Work out how long she has to wait. h Answer (c)(ii) min [1] (d) The plane takes off at 12 40 Spanish time, which is 11 40 English time. The flight takes 2 hours. What is the time in England when she arrives? Answer (d) [1] (e) The plane has seats for 420 passengers. 15% of the seats are empty. How many passengers are on the plane? Answer (e) [3] 4 1

3 © UCLES 2005 0580/03, 0581/03 Jun 05 [Turn over For Examiner's Use 2 (a) Complete the table of values for y = 1 + 2x – x2. x 3 − 2 − 1 − 0 1 2 3 4 5 y 14 − 7 − 1 2 − 14 − [3] (b) Draw the graph of y = 1 + 2x – x2 on the grid below. y x 4 2 –2 –4 –6 –8 –10 –12 –14 0 1 5 4 3 2 –3 –2 –1 [4] (c) Use your graph to find the solutions to the equation 1 + 2x – x2 = 0. Answer (c) x = or x = [2] (d) (i) On the grid, draw the line of symmetry of the graph. [1] (ii) Write down the equation of this line of symmetry. Answer(d)(ii) [1]

4 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use 3 Monday Tuesday Wednesday Thursday Friday Saturday Sunday Minimum temperature oC 4 6 0 −2 −4 2 Maximum temperature oC 8 10 5 7 2 7 The table shows the minimum and maximum temperatures on six days of a week. (a) (i) On Sunday the minimum temperature was 5 °C lower than on Saturday. The maximum temperature was 2 °C higher than on Saturday. Use this information to complete the table. [2] (ii) Find the difference between the minimum and maximum temperatures on Thursday. Answer(a)(ii) oC [1] (b) Use the table to complete the graphs below for all seven days. Temperature oC 10 8 6 4 2 0 –2 –4 –6 Monday Tuesday Wednesday Thursday Friday Saturday Sunday Minimum temperature Maximum temperature [2]

5 © UCLES 2005 0580/03, 0581/03 Jun 05 [Turn over For Examiner's Use (c) Use your graphs to find (i) on how many days the temperature fell below −1 °C, Answer(c)(i) [1] (ii) which day had the largest difference between minimum and maximum temperatures. Answer(c)(ii) [1] (d) The formula for changing degrees Celsius (C) to degrees Fahrenheit (F) is 32 5 9 + = C F . Use the formula to change 6 degrees Celsius to degrees Fahrenheit. Show all your working. Answer(d) [2]

6 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use 4 0 –5 –4 –3 –2 –1 4 3 2 1 –1 –2 –3 –4 1 2 3 4 5 6 A C D B y x (a) A translation is given by 3 6 + 4 3 − − . (i) Write this translation as a single column vector. Answer(a)(i) [2] (ii) On the grid, draw the translation of triangle A using this vector. [2] (b) Another translation is given by –2 1 1 − (i) Write this translation as a single column vector. Answer(b)(i) [2] (ii) On the grid, draw the translation of triangle B using this vector. [2] (c) Describe fully the single transformation that maps shape C onto shape D. Answer(c) [3]

7 © UCLES 2005 0580/03, 0581/03 Jun 05 [Turn over For Examiner's Use (d) y x O The triangle in the diagram above is isosceles. (i) How many lines of symmetry does this triangle have? Answer(d)(i) [1] (ii) Write down the order of rotational symmetry of this triangle. Answer(d)(ii) [1] (iii) On the grid above, draw the rotation of this triangle about O through 180o. [2] (iv) Describe fully another single transformation that maps this triangle onto your answer for part (d)(iii). Answer(d)(iv) [2]

8 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use 5 6 3 5 1 4 2 (a) Asif tests a six-sided spinner. The results of 60 spins are shown below. 3 3 6 5 6 1 2 6 5 2 3 4 4 4 3 4 6 5 2 1 6 3 6 4 1 5 3 6 2 6 6 6 3 6 1 6 6 5 1 6 1 6 2 5 3 6 4 2 3 5 1 4 4 1 5 4 6 6 2 3 (i) Use these results to complete the frequency table. Number Frequency 1 2 3 4 5 6 [3] (ii) Write down the mode. Answer(a)(ii) [1] (iii) Find the median. Answer(a)(iii) [2]

9 © UCLES 2005 0580/03, 0581/03 Jun 05 [Turn over For Examiner's Use (iv) Calculate the mean. Give your answer correct to one decimal place. Answer(a)(iv) [3] (b) Asif tests a different six-sided spinner. He draws a bar chart to show the results. 14 12 10 8 6 4 2 0 1 2 3 4 5 6 Number Frequency (i) How many times did he spin this spinner? Answer(b)(i) [2] (ii) Calculate the mean score for this spinner. Answer(b)(ii) [3]

10 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use 6 (a) x cm 2x cm NOT TO SCALE The perimeter of the rectangle in the diagram above is 36 centimetres. (i) Find the value of x. Answer(a)(i) x = [2] (ii) Using this value of x, calculate the area of the rectangle. Answer(a)(ii) cm2 [2] (b) 4z + 2 10z – 1 y + 3 3y NOT TO SCALE The diagram above shows another rectangle. (i) In this rectangle 3y = y + 3. Solve the equation to find y. Answer(b)(i) y = [2] (ii) Write down an equation in z. Answer(b)(ii) [1] (iii) Solve the equation in part (b)(ii) to find z. Answer(b)(iii) z = [3]

11 © UCLES 2005 0580/03, 0581/03 Jun 05 [Turn over For Examiner's Use (c) 17 3 4a+b a–b NOT TO SCALE The diagram above shows another rectangle. (i) Write down two equations in a and b. Answer(c)(i) [2] (ii) Solve these two equations simultaneously to find a and b. Answer(c)(ii) a = b = [3]

12 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use 7 North North 14 km A B C R Land Sea At midday, a ship is somewhere along the line from A to C. (a) By measuring an angle, write down the three figure bearing of the ship from A. Answer(a) [2] (b) The coastguard at B sees the ship on a bearing of 350o. (i) On the diagram draw accurately the line showing a bearing of 350o from B. [1] (ii) On the diagram mark the position of the ship, S. [1] (c) (i) Measure the length, in centimetres, of the line AB on the diagram. Answer(c)(i) cm [1] (ii) The distance from A to B is 14 kilometres. Calculate the scale of the drawing. Give your answer in the form 1:n. Answer(c)(ii) 1: [2]

13 © UCLES 2005 0580/03, 0581/03 Jun 05 [Turn over For Examiner's Use (d) The ship is sailing straight for the rocks, R. There is a lighthouse at A. The range of the light from the lighthouse is 10 kilometres. (i) Using your scale, draw the locus of points that are 10 kilometres from A. [2] (ii) Draw the line SR on the diagram. How far is the ship from the rocks when the light from the lighthouse is first seen on the ship? Answer(d)(ii) km [2] (e) If the ship does not alter course it will hit the rocks at 12 40. A lifeboat sets off from the coastguard station, B, at 12 00 and sails straight towards the rocks. (i) Measure and calculate the distance, in kilometres, from the coastguard station, B, to the rocks, R. Answer(e)(i) km [2] (ii) Calculate the speed, in kilometres per hour, at which the lifeboat must sail to reach the rocks by 12 40. Answer(e)(ii) km/h [3] (iii) A knot is 1 nautical mile per hour. One nautical mile is equal to 1.85 kilometres. Calculate the speed found in part (e)(ii) in knots. Answer(e)(iii) knots [2]

14 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use 8 A A B B C C NOT TO SCALE NOT TO SCALE 4 cm 6 cm 8 cm The diagram above shows a cuboid and its net. (a) Calculate the total surface area of the cuboid. Answer(a) cm2 [3]

15 © UCLES 2005 0580/03, 0581/03 Jun 05 For Examiner's Use (b) Calculate the volume of the cuboid. Answer(b) cm3 [2] (c) An ant walks directly from A to C on the surface of the cuboid. (i) Draw a straight line on the net to show this route. [1] (ii) Calculate the length of the ant’s journey. Answer(c)(ii) cm [3] (iii) Calculate the size of angle CAB on the net. Answer(c)(iii) Angle CAB = [3]

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 University of Cambridge Local Examinations Syndicate (UCLES), which is itself a department of the University of Cambridge. 0580/03, 0581/03 Jun 05 BLANK PAGE

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the June 2005 question paper 0580/0581 MATHEMATICS 0580/03, 0581/03 Paper 3 (Core), maximum raw mark 104 This mark scheme is published as an aid to teachers and students, to indicate the requirements of the examination. It shows the basis on which Examiners were initially instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began. Any substantial changes to the mark scheme that arose from these discussions will be recorded in the published Report on the Examination. All Examiners are instructed that alternative correct answers and unexpected approaches in candidates’ scripts must be given marks that fairly reflect the relevant knowledge and skills demonstrated. Mark schemes must be read in conjunction with the question papers and the Report on the Examination. • CIE will not enter into discussion or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the June 2005 question papers for most IGCSE and GCE Advanced Level and Advanced Subsidiary Level syllabuses and some Ordinary Level syllabuses www.XtremePapers.com

Grade thresholds for Syllabus 0580/0581 (Mathematics) in the June 2005 examination. minimum mark required for grade: maximum mark available A C E F Component 3 104 N/A 76 50 39 The threshold (minimum mark) for B is set halfway between those for Grades A and C. The threshold (minimum mark) for D is set halfway between those for Grades C and E. The threshold (minimum mark) for G is set as many marks below the F threshold as the E threshold is above it. Grade A* does not exist at the level of an individual component.

June 2005 IGCSE MARK SCHEME MAXIMUM MARK: 104 SYLLABUS/COMPONENT: 0580/03, 0581/03 MATHEMATICS Paper 3 (Core)

TYPES OF MARK Most of the marks (those without prefixes, and ‘B’ marks) are given for accurate results, drawings or statements. • M marks are given for a correct method. • B marks are given for a correct statement or step. • A marks are given for an accurate answer following a correct method. ABBREVIATIONS a.r.t. Anything rounding to b.o.d. Benefit of the doubt has been given to the candidate c.a.o. Correct answer only (i.e. no ‘follow through’) e.e.o. Each error or omission f.t. Follow through i.s.w. Ignore subsequent working o.e. Or equivalent SC Special case s.o.i. Seen or implied ww Without working www Without wrong working Work followed through after an error: no further error made

Page 1 Mark Scheme Syllabus Paper IGCSE – JUNE 2005 0580/0581 3 © University of Cambridge International Examinations 2005 Question Answer Marks Comments 1 (a) 2.8 1 ignore minus sign, accept 2800 g (b) 106.5(0) 1 107 is X (but remember to look back for 106.5) (c) (i) 10 40 1 accept 10.40, 10:40, 10.40 am (ii) 1 (hour) 30 (mins) 1 f.t. f.t. from (c)(i) [f.t. is (c)(i) > 12 10] accept 1 ½ (hours), 1.5 (hours), 90 (mins) (d) 13.55 1 accept 1.55 (pm) but 01 55 and 1.55 am are X (e) 357 3 M2 for 420 – 15 x 420/100, 420 x 85/100 o.e. or M1 for 15 x 420/100 o.e. answer of 63 is M1 implied 8 2 (a) –2 1 2 –7 3 B2 for 3 correct, B1 for 1 or 2 correct (b) 9 correct points plotted P3 f.t. P2 f.t. for 7 or 8 correct, P1 f.t. for 5 or 6 correct limit for acurracy is ½ small square smooth curve drawn C1 must go through the 9 correct points not dependent on P3 (c) –0.4 ( ± /0.1) 1 please note no f.t. on this part 2.4 ( ± 0.1) 1 (d) (i) correct line drawn 1 accept dotted/dashed line must be full length from (1, –14) to (1,2) (ii) x = 1 1 f.t. f.t. from (d)(i) if x = k any reference to y is X 11 3 (a) (i) –3 9 1 1 (ii) 9 1 ignore minus sign (b) correct max drawn correct min drawn 1 f.t. 1 f.t. } f.t. is from (a)(i) [Sunday] } allow Sunday (only) to be 1 square out horizontally } allow freehand straight lines

Page 2 Mark Scheme Syllabus Paper IGCSE – JUNE 2005 0580/0581 3 © University of Cambridge International Examinations 2005 (c) (i) 3 1 f.t. f.t. is 3 if Sunday negative otherwise 2 allow 3 out of 7 (ii) Sunday 1 f.t. f.t. if not Sunday is Thursday (d) 42.8 2 M1 for 9 x 6/5 + 32 or better e.g. 54/5 + 32, 10.8 + 32 answer of 43 is M1 implied 9 4 (a) (i) 3 –1 1 1 (ii) correct translation drawn 1 f.t. } f.t. where possible (i.e. still on the grid) 1 f.t. } condone inaccuracy/unruled if intention is clear } if ½ scale used then penalise first occurence only (–1) (b) (i) –2 2 1 1 (ii) correct translation drawn 1 f.t. } f.t. where possible (i.e. still on the grid) 1 f.t. } condone inaccuracy/unruled if intention is clear (c) enlargement (centre) (0,0) o.e. (scale factor) 2 1 1 1 } } must be a single transformation } (d) (i) 1 1 (ii) 1 1 (iii) correct rotation drawn 2 SC1 for 180 rotation about any other point SC1 for ± 90 rotation about O (iv) reflection in the x-axis oe M1 B1(dep) } must be a single transformation } condone inaccuracy/unruled if intention is clear } enlargement, s.f. = –1, centre (0,0) is B2 17 5 (a) (i) 8 7 10 9 8 18 3 2 for 4 or 5 correct, 1 for 2 or 3 correct accept tallies if in 5’s, accept 8/60, 7/60 etc. (ii) 6 1 c.a.o (iii) 4 2 c.a.o M1 for evidence of ranking (cum. freq.)

Page 3 Mark Scheme Syllabus Paper IGCSE – JUNE 2005 0580/0581 3 © University of Cambridge International Examinations 2005 (iv) 3.9 3 c.a.o M1 (f.t.) for 8 x 1 + 7 x 2 + 10 x 3 or 8 +14 +30 (min 3) M1 (f.t.) dep. for /60 [both M marks may be by the table] answer of 3.93(3333) is M2 implied 39.3(33...) is M1 implied (b) (i) 60 2 M1 for 10 + 7 + 10 + 7 + 14 + 12 (min 3) (ii) 3.7(3333 ) 3 M1 (f.t.) for 10 x 1 + 7 x 2 + 10 x 3… or 10 +14 + 30… (min 3) M1 (f.t.) dep. for /(b)(i) 14 6 (a) (i) 6 2 M1 for 6x = 36 or 3x = 18 o.e. (ii) 72 2 f.t. f.t. is 2 x (a)(i) x (a)(i) M1 (f.t.) for 6 x 12, 2 x 36, 2 x 6 x 6 (b) (i) 1.5 or 1 ½ or 3/2 2 M1 for 3y – y = 3 o.e. [unknown on one side] (ii) 4z + 2 = 10z – 1 1 accept any equivalent equation in z if (b)(ii) is left blank may recover mark if 4z + 2 = 10z – 1 seen in (b)(iii) (iii) 0.5 or ½ or 3/6 3 B1 for correct single z term B1 for correct single constant term (c) (i) a – b = 3 o.e. 4a + b = 17 o.e. 5a = 20 4a + b + 3 = a – b + 17 } 1,1 } if (c)(i) is left blank may recover mark(s) with a – b = 3, 4a + b = 17, 5a = 20 seen in (c)(ii) (ii) (a=) 4 and (b=) 1 3 2 for either (a=) 4 or (b=) 1 or M1 (f.t.) for correctly eliminating one of the variables 15 7 (a) 050 ( ± 2) 2 M1 for correct angle but not 3 figures i.e. 50 ( ) 2 ± (b) (i) correct line drawn ( ± 2) 1 length at least 3 cms long (ii) correct position marked 1 f.t. f.t is from line drawn in (b)(i) ( 2 ± mm) but must be on the line AC (c) (i) 7 ( ± 2 mm) 1 (ii) 200000 2 c.a.o. 1 for figs 2 or SC1 for figs 1.94 to 2.06

Page 4 Mark Scheme Syllabus Paper IGCSE – JUNE 2005 0580/0581 3 © University of Cambridge International Examinations 2005 (d) (i) correct locus drawn 2 f.t. f.t. is for their scale (normally 5 cm) at least over sea allow dotted/dashed locus SC1 for any other circle with centre A drawn SC1 for ¼ correct circle over sea (ii) correct line SR drawn 5 to 6 incl. 1 f.t. 1 f.t. is for their S allow dotted/dashed line no f.t. on this part (e) (i) 18.6 to 19.4 incl. 2 SC1 for 9.3 to 9.7 incl. seen (ii) 27.9 to 29.1 incl. 3 M1 for conversion of minutes to hours (min of 0.66, 0.67 if dec.) M1 (indep) f.t. for their distance (e)(i)/their time taken (iii) 15.4 2 f.t. f.t. is (e)(ii)/1.85 M1 for (e)(ii)/1.85 seen 18 8 (a) 208 3 M2 for 2(24 + 32 + 48) or 48 + 64 + 96 or 160 + 24 + 24 o.e. or M1 for 24 or 32 or 48 or 160 seen (b) 192 2 M1 for 6 x 8 x 4 (c) (i) straight line AC 1 (ii) 12.8 3 M2 for 10 + 8 or 100 + 64 or 164 or M1 for 10 + 8 or 100 + 64 or 164 or SC1 for complete correct use of Pythagoras (iii) 51.3 or 51.4 3 M1 for 10/8 and tan seen o.e. and M1 for tan 10/8 seen o.e. [the o.e include sin or cos with their (c)(ii)] or SC1 for complete correct use of a trig. ratio 12 104