Cambridge IGCSE Mathematics 0580 — 2005 May/June Paper 2 · Variant 1

This document consists of 11 printed pages and 1 blank page. IB05 06_0580_02/4RP  UCLES 2005 [Turn over UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS Paper 2 (Extended) 0580/02 0581/02 Candidates answer on the Question Paper. Additional Materials: Electronic calculator Geometrical instruments May/June 2005 Mathematical tables (optional) Tracing paper (optional) 1hour 30 minutes 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 70. 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 *058002* For Examiner's Use www.XtremePapers.com

2 © UCLES 2005 0580/02, 0581/02 Jun 05 For Examiner's Use 1 Calculate 52 25 (a) giving your answer as a fraction, Answer (a) [1] (b) giving your answer as a decimal. Answer (b) [1] 2 NOT TO SCALE pavement entrance 5o 3.17 m h m A shop has a wheelchair ramp to its entrance from the pavement. The ramp is 3.17 metres long and is inclined at 5o to the horizontal. Calculate the height, h metres, of the entrance above the pavement. Show all your working. Answer m [2] 3 A block of cheese, of mass 8 kilograms, is cut by a machine into 500 equal slices. (a) Calculate the mass of one slice of cheese in kilograms. Answer (a) kg [1] (b) Write your answer to part (a) in standard form. Answer (b) kg [1]

3 © UCLES 2005 0580/02, 0581/02 Jun 05 [Turn over For Examiner's Use 4 Calculate the value of (cos 40o)2 + (sin 40o)2. Answer [2] 5 (a) Write down the order of rotational symmetry of the diagram. Answer (a) [1] (b) Draw the lines of symmetry on the diagram. [1] 6 A square ABCD, of side 8 cm, has another square, PQRS, drawn inside it. P,Q,R and S are at the midpoints of each side of the square ABCD, as shown in the diagram. A B D C P S R Q NOT TO SCALE (a) Calculate the length of PQ. Answer (a) cm [2] (b) Calculate the area of the square PQRS. Answer (b) cm2 [1]

4 © UCLES 2005 0580/02, 0581/02 Jun 05 For Examiner's Use 7 To raise money for charity, Jalaj walks 22 km, correct to the nearest kilometre, every day for 5 days. (a) Complete the statement in the answer space for the distance, d km, he walks in one day. d < Answer (a) [2] (b) He raises $1.60 for every kilometre that he walks. Calculate the least amount of money that he raises at the end of the 5 days. Answer (b) $ [1] 8 Solve the simultaneous equations 2 1 x + 2y = 16, 2x + 2 1 y = 19. Answer x = y = [3] 9 The wavelength, w, of a radio signal is inversely proportional to its frequency, f. When f = 200, w = 1500. (a) Find an equation connecting f and w. Answer (a) [2] (b) Find the value of f when w = 600. Answer (b) f = [1]

5 © UCLES 2005 0580/02, 0581/02 Jun 05 [Turn over For Examiner's Use 10 Rooms in a hotel are numbered from 1 to 19. Rooms are allocated at random as guests arrive. (a) What is the probability that the first guest to arrive is given a room which is a prime number? (1 is not a prime number.) Answer (a) [2] (b) The first guest to arrive is given a room which is a prime number. What is the probability that the second guest to arrive is given a room which is a prime number? Answer (b) [1] 11 n( ) = 21, n(A∪B) = 19, n(A∩B' ) = 8 and n(A) = 12. Complete the Venn diagram to show this information. Answer … … … … A B [3] 12 M = x x x x 2 2 . Find (a) 2M, Answer (a) [1] (b) M2. Answer (b) [2]

6 © UCLES 2005 0580/02, 0581/02 Jun 05 For Examiner's Use 13 Pattern 1 Pattern 2 Pattern 3 The first three patterns in a sequence are shown above. (a) Complete the table. Pattern number 1 2 3 4 Number of dots 5 [1] (b) Find a formula for the number of dots, d, in the nth pattern. Answer (b) d= [1] (c) Find the number of dots in the 60th pattern. Answer (c) [1] (d) Find the number of the pattern that has 89 dots. Answer (d) [1] 14 A house was built in 1985 and cost $62 000. It was sold in 2003 for $310 000. (a) Work out the 1985 price as a percentage of the 2003 price. Answer (a) % [2] (b) Calculate the percentage increase in the price from 1985 to 2003. Answer (b) % [2]

7 © UCLES 2005 0580/02, 0581/02 Jun 05 [Turn over For Examiner's Use 15 The points A(6,2) and B(8,5) lie on a straight line. (a) Work out the gradient of this line. Answer (a) [1] (b) Work out the equation of the line, giving your answer in the form y = mx + c. Answer (b) [2] 16 Simplify 2 2 + − + x x x x . Write your answer as a fraction in its simplest form. Answer [3]

8 © UCLES 2005 0580/02, 0581/02 Jun 05 For Examiner's Use 17 mark water The height, h metres, of the water, above a mark on a harbour wall, changes with the tide. It is given by the equation h = 3sin(30t)o where t is the time in hours after midday. (a) Calculate the value of h at midday. Answer (a) [1] (b) Calculate the value of h at 19 00. Answer (b) [2] (c) Explain the meaning of the negative sign in your answer. Answer (c) [1]

9 © UCLES 2005 0580/02, 0581/02 Jun 05 [Turn over For Examiner's Use 18 Revina has to pass a written test and a driving test before she can drive a car on her own. The probability that she passes the written test is 0.6. The probability that she passes the driving test is 0.7. (a) Complete the tree diagram below. Pass Fail Pass Fail Pass Fail 0.7 … 0.6 … 0.7 … Written test Driving test [1] (b) Calculate the probability that Revina passes only one of the two tests. Answer (b) [3] 19 Solve (a) 0.2x + 3.6 = 1.2, Answer (a) x= [2] (b) 2 – 3x < x + 2. 5 Answer (b) [3]

10 © UCLES 2005 0580/02, 0581/02 Jun 05 For Examiner's Use 20 A plane flies from Auckland (A) to Gisborne (G) on a bearing of 115o. The plane then flies on to Wellington (W). Angle AGW = 63o. 115o 63o A G North North W 400 km 410 km NOT TO SCALE (a) Calculate the bearing of Wellington from Gisborne. Answer (a) [2] (b) The distance from Wellington to Gisborne is 400 kilometres. The distance from Auckland to Wellington is 410 kilometres. Calculate the bearing of Wellington from Auckland. Answer (b) [4]

11 © UCLES 2005 0580/02, 0581/02 Jun 05 For Examiner's Use 21 40o 5 cm A B C O NOT TO SCALE A, B and C are points on a circle, centre O. Angle AOB = 40o. (a) (i) Write down the size of angle ACB. Answer (a)(i) Angle ACB = [1] (ii) Find the size of angle OAB. Answer (a)(ii) Angle OAB = [1] (b) The radius of the circle is 5 cm. (i) Calculate the length of the minor arc AB. Answer (b)(i) cm[2] (ii) Calculate the area of the minor sector OAB. Answer (b)(ii) cm2 [2]

12 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. 0580/02, 0581/02 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 0580/02, 0581/02 Paper 2 (Extended), maximum raw mark 70 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 initialy 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 2 70 58 36 23 n/a 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.

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

June 2005 IGCSE MARK SCHEME MAXIMUM MARK: 70 SYLLABUS/COMPONENT: 0580/02, 0581/02 MATHEMATICS Paper 2 (Extended)

Page 1 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS – JUNE 2005 0580/0581 2 © University of Cambridge International Examinations 2005 * indicates that it is necessary to look in the working following a wrong answer 1 (a) (b) 25/32 0.781 (25) 1 1√ 2 0.276 2* M1 sin 5° = h/3.17 0.28 may score M0 3 (a) (b) 0.016 1.6 × 10 -2 1 1√ Allow 2/125 x 10 essential 4 1(.00) or 0.9r 2 M1 A0 other answers in range 0.99 to 1.053 5 (a) (b) 3 3 lines 1 1 by eye 6 (a) (b) 5.66 32(.0) 2 1√ M1 42 + 42 or 4/sin45 or 4√2 or √32 (a)2 from the answer space 7 (a) (b) 21.5 22.5 172 1,1 1√ SC1 correct but reversed (a) least value x 8 8 x = 8 y = 6 3* M1 for multiplication and subtraction 9 (a) (b) wf = 300000 oe 500 2 1√ M1 wf = k A1 k = 300000 10 (a) (b) 8/19 or 0.421 7/18 or 0.389 2 1√ M1 their prime number count/19 11 3 B1 for 8 in correct place B1 for 2 in correct place B1 for 4 and 7 in correct place SC2 2 4 8 7 or 2 6 6 7 12 (a) (b)     x x x x 2 4 4 2         2 2 2 2 5 4 4 5 x x x x 1 2* M1         + + + + 2 2 2 2 2 2 2 2 4 2 2 2 2 4 x x x x x x x x 13 (a) (b) (c) (d) 8, 11, 14 3n + 2 182 29 1 1 1√ 1√ integers only 14 (a) (b) 20% 400% 2* 2* M1 for 310000 62000 x 100 M1 for 62000 248000 x 100 15 (a) (b) 3/2 oe y = 3/2x – 7 1 2*√ M1 correct method 16 ) 2 ( )1 ( 4 + + x x x oe 3* M1 (x + 2)(x +2) - x2 B1 x2 + 4x + 4

Page 2 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS – JUNE 2005 0580/0581 2 © University of Cambridge International Examinations 2005 17 (a) (b) (c) 0 –1.5 below the height at midday 1 2* 1 M1 for t = 7 18 (a) (b) 0.4,0.3, 0.3 0.46 1 3* M1 0.6 × “0.3” or “0.4” × 0.7 dep M1 add 19 (a) –12 x > –1 cao 2* 3* M1 0.2x = –2.4 B1 for every two moves completed correctly 20 (a) (b) 232o 175(.4)° 2* 4* M1 for 360 – (63 + “65”) M1 for 63 sin 410 = x sin 400 A1 GAW = 60.4 M1 115 + GAW and no further working A1 √ 21 (a) (b) (i) (ii) (i) (ii) 20 70 3.49 8.73 1 1 2* 2* M1 360 40 x 2 x π x 5 M1 360 40 x π x 52 TOTAL 70