Cambridge IGCSE Mathematics (with coursework) 0581 — 2004 Oct/Nov Paper 3 · Variant 1

This document consists of 15 printed pages and 1 blank page. IB04 11_0580_03/4RP © UCLES 2004 [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 October/November 2004 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 *058001* For Examiner's Use www.XtremePapers.com

2 © UCLES 2004 0580/03/O/N/04 For Examiner's Use 1 (a) Two friends, Hatab and Yasin, went on a cycle ride. Part of the distance-time graph for their journey is shown below. 10 00 18 16 14 12 10 8 6 4 2 0 11 00 12 00 13 00 14 00 15 00 Hatab and Yasin Yasin Hatab Time of day Distance from home (km) For the first part of the journey they cycled at the same speed. (i) Find their speed for the first part of the journey. Answer(a)(i) km/h [1] (ii) At 11 00 they stopped for half an hour. Show this on the graph. [1] (iii) They continued on their ride and at 12 45 they were 16 kilometres from home. Show this part of the journey on the graph. [1] (iv) They stopped again and then had a race going home. (a) For how long did they stop? Answer(a)(iv)(a) min [1] (b) Who won the race? Answer(a)(iv)(b) [1] (v) What was the total length of their journey? Answer(a)(v) km [1]

3 © UCLES 2004 0580/03/O/N/04 [Turn over For Examiner's Use (b) On a certain day the conversion rate between dollars ($) and Indian rupees was $1 = 45 rupees. (i) How many rupees were equivalent to $10? Answer(b)(i) rupees [1] (ii) Use this information to draw a conversion graph on the axes below. 500 400 300 200 100 0 1 2 3 4 5 6 7 8 9 10 11 Dollars ($) Rupees [2] (iii) Use your graph to find (a) how many rupees were equivalent to $6.80, Answer(b)(iii)(a) rupees [1] (b) how many dollars were equivalent to 480 rupees. Answer(b)(iii)(b) $ [1]

4 © UCLES 2004 0580/03/O/N/04 For Examiner's Use 2 0 1 _1 _2 _3 _4 _5 _6 _7 2 3 4 5 6 7 x y A B C D 6 5 4 3 2 1 _1 _2 _3 _4 _5 _6 (a) Describe fully the single transformation that maps triangle A onto triangle B. Answer(a) [3] (b) Describe fully the single transformation that maps triangle A onto triangle C. Answer(b) [3] (c) Find the centre and the scale factor of the enlargement that maps triangle A onto triangle D. ( , ) scale factor Answer(c) centre [2] (d) On the grid (i) draw the image of triangle A under a reflection in the line x = −1, [2] (ii) draw the image of triangle B under a rotation of 180° about (−4, −3). [2]

5 © UCLES 2004 0580/03/O/N/04 [Turn over For Examiner's Use 3 40o A E C D B 6 cm 2 cm 10 cm NOT TO SCALE On the above diagram, AB = 2 cm, BD = 6 cm, AE = 10 cm, angle BCD = 40° and angle BDE = 90°. (a) Write down the length of AD. Answer(a) AD = cm [1] (b) Calculate the length of DE. Answer(b) DE = cm [2] (c) Calculate the size of angle AED. Answer(c) angle AED = [2] (d) Calculate the length of CD. Answer(d) CD = cm [3] (e) Find the length of CE. Answer(e) CE = cm [1]

6 © UCLES 2004 0580/03/O/N/04 For Examiner's Use 4 (a) A B C 5 cm 6 cm 4 cm NOT TO SCALE (i) In the space below, using a ruler and compasses only, construct the above triangle accurately. [3] (ii) Using the triangle you have drawn, measure and write down the size of angle ACB. Answer(a)(ii) angle ACB = [1]

7 © UCLES 2004 0580/03/O/N/04 [Turn over For Examiner's Use (b) In the diagram below two points, P and Q, are joined by a straight line. P Q (i) On the diagram draw the locus of all the points that are 4 centimetres from the line PQ. [3] (ii) On the same diagram, using a straight edge and compasses only, construct the locus of the points that are equidistant from P and Q. Show all your construction lines. [2] (iii) Shade the region which contains the points that are closer to P than to Q and are less than 4 centimetres from the line PQ. [2]

8 © UCLES 2004 0580/03/O/N/04 For Examiner's Use 5 (a) 80o yo 140o A C B D NOT TO SCALE In the diagram above AB=BC and AD=DC. (i) What is the special name of the quadrilateral ABCD? Answer(a)(i) [1] (ii) On the diagram draw the line of symmetry. [1] (iii) Calculate the value of y. Answer(a)(iii) y = [2] (b) N M L K O 40o ro po qo NOT TO SCALE In the diagram above, the points K,L,M and N lie on the circle centre O. KN is parallel to LM. Find the values of p,q and r. p = , q = , r = [3] Answer(b)

9 © UCLES 2004 0580/03/O/N/04 [Turn over For Examiner's Use (c) xo NOT TO SCALE The diagram above shows a regular seven-sided polygon. Each of the interior angles measures x°. One of the angles is marked in the diagram. Calculate the value of x, giving your answer correct to 1 decimal place. Show all your working. Answer(c) x = [4]

10 © UCLES 2004 0580/03/O/N/04 For Examiner's Use 6 (a) Complete the table below for y = x2 − 2x. x −2 −1 0 1 2 3 4 y 8 −1 3 8 [3] (b) On the grid below, draw the graph of y = x2 − 2x for −2 x 4. 4 2 1 _1 0 8 7 6 5 4 3 2 1 _1 _2 y 3 _2 _3 _4 x y = 2 [4] (c) The line y = 2 is drawn on the diagram. Use your graph to find the values of x that solve the equation x2 − 2x = 2. x = or x = Answer(c) [2] (d) Complete the table below for y = 4 − x. x −4 0 4 y 8 [2] (e) On the grid above, draw the line y = 4 − x for −4 x 4. [1] (f) Write down the x coordinates of the points of intersection of the graphs of y = x2 − 2x and y = 4 − x. x = or x = Answer(f) [2]

11 © UCLES 2004 0580/03/O/N/04 [Turn over For Examiner's Use 7 (a) Rajeesh thought of a number. He multiplied this number by 2. He then added 10. The answer was 42. (i) What was the number Rajeesh first thought of? Answer(a)(i) [1] (ii) Simon thought of a number x. He multiplied this number by 3 and then added 8. Write down an expression in x for his answer. Answer(a)(ii) [2] (b) Simplify − 8a + 7b − a − 2b. Answer(b) [2] (c) Factorise fully 6a − 9a2 . Answer(c) [2] (d) Make t the subject of the formula v = u + at. Answer(d) t= [2] (e) Solve the simultaneous equations 8x + 2y = 13, 3x + y = 4. x = , y = Answer(e) [4]

12 © UCLES 2004 0580/03/O/N/04 For Examiner's Use 8 (a) The list shows the rainfall in millimetres in Prestbury for the 12 months of 2002. 61 146 22 54 67 94 141 22 37 167 87 170 (i) Write down the mode. Answer(a)(i) mm [1] (ii) Find the median. Answer(a)(ii) mm [2] (iii) Calculate the mean. Answer(a)(iii) mm [2] (b) During the years 1996 - 2000 the total rainfall in Prestbury was 5400 millimetres. The pie chart shows how this was spread over the five years. 1996 1997 1998 1999 2000

13 © UCLES 2004 0580/03/O/N/04 [Turn over For Examiner's Use (i) Measure the angles of the sectors for 1998, 1999 and 2000. Write your answers in the table below. [3] (ii) Work out the annual rainfall, in millimetres, for each of the years 1998, 1999 and 2000. Write your answers in the table below. [3] Answers (b)(i) and (ii) Year Angle (degrees) Rainfall (mm) 1996 54 810 1997 60 900 1998 1999 2000 Total 360 5400 (iii) What do you notice about the trend in the rainfall from 1996 to 2000? Answer(b)(iii) [1]

14 © UCLES 2004 0580/03/O/N/04 For Examiner's Use 9 (a) A pattern of numbers is shown below. 26 1 2 3 4 5 6 17 … 10 18 … 5 11 19 … 2 6 12 20 … 1 3 7 13 21 … 4 8 14 22 … 9 15 23 … 16 24 … 25 … … row (i) On the diagram complete row 6. [1] (ii) The last numbers in each row form a sequence. 1, 4, 9, 16, 25, …………… (a) What is the special name given to these numbers? Answer(a)(ii)(a) [1] (b) Write down the last number in the 10th row. Answer(a)(ii)(b) [1] (c) Write down an expression for the last number in the nth row. Answer(a)(ii)(c) [1] (iii) The numbers in the middle column of the pattern form a sequence. 1, 3, 7, 13, 21, 31, ………….. (a) Write down the next number in this sequence. Answer(a)(iii)(a) [1] (b) The expression for the nth number in this sequence is n2 − n + 1. Work out the 30th number. Answer(a)(iii)(b) [2]

15 © UCLES 2004 0580/03/O/N/04 For Examiner's Use (b) Another pattern of numbers is shown below. row 1 1 2 3 4 5 6 7 8 9 10 2 11 12 13 14 15 16 17 18 19 20 3 21 22 23 24 25 26 27 28 29 30 4 31 32 33 34 35 36 37 38 39 40 (i) What is the last number in the 10th row? Answer(b)(i) [1] (ii) Find an expression for the last number in the nth row. Answer(b)(ii) [1] (iii) What is the first number in the 10th row? Answer(b)(iii) [1] (iv) Find an expression for the first number in the nth row. Answer(b)(iv) [1]

16 Every reasonable effort has been made to trace all copyright holders where the publishers (i.e. UCLES) are aware that third-party material has been reproduced. The publishers would be pleased to hear from anyone whose rights they have unwittingly infringed. 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/O/N/04 BLANK PAGE

CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the November 2004 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 November 2004 question papers for most IGCSE and GCE Advanced Level syllabuses. www.XtremePapers.com

Grade thresholds taken for Syllabus 0580/0581 (Mathematics) in the November 2004 examination. minimum mark required for grade: maximum mark available A C E F Component 3 104 N/A 78 55 45 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 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

November 2004 INTERNATIONAL GCSE MARK SCHEME MAXIMUM MARK: 104 SYLLABUS/COMPONENT: 0580/03, 0581/03 MATHEMATICS Paper 3

Page 1 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 © University of Cambridge International Examinations 2005 Question number Mark Scheme Part Marks Notes Question Total 1 a) i) 10 1 ii) straight line from (11,10) to (11 30,10) 1 iii) straight line from (11 30,10) to (12 45,16) 1√ allow +2 mm in length by eye but must go through the correct points. f.t. from their (1130,10) iv) a) 15 1 allow ¼ hour b) Hatab 1 v) 32 1 b) i) 450 1 ii) straight line ruled from (1,45) to (10,450) 2 SC1 for freehand or broken line or any straight line through the origin ± ½ small square at both points iii) a) 306 ± 4 1 b) 10 60 to 10.80 1 allow 10.6 etc. 11 2 a) translation 1 must be single transformation       − − 7 6 1 1 SC1 for correct vector inverted, or       − − 14 12 , or for correct row vector, or co-ordinates. Condone missing brackets b) rotation M1 must be single transformation -90 or 90 clockwise o.e. A1 about (0, 0) o.e. A1

Page 2 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 © University of Cambridge International Examinations 2005 c) (0, 0) 1 1.5 o.e. 1 not 3:2 etc. d) i) correct triangle drawn 2 SC1 for reflection of A in any vertical line or in y = -1 ii) correct triangle drawn 2 SC1 for 180o rotation about any point or SC1 for rotation ± 90o about (-4,-3) 12 3 In this question alternative methods must be complete a) 8 1 b) 6 2 M1 for 64 100 − o.e. must show square root c) art 53.1 2 M1 for sin and 8/10 seen o.e. d) art 7.15 3 M1 for tan 40 and 6 seen +M1 for 6/tan 40 o.e. e) 13.15 or 13.2 1√ f.t. for their b) + d) to 3 s.f. or better 9 4 a) i) triangle drawn with three sides the correct length ± 0.1 cm 3 2 for two sides correct, with arcs 1 for two sides correct without arcs ii) 56 ± 2 c.a.o. 1 b) in this part of the question deduct 1 once for broken lines i) complete locus drawn 3 1 for a line correct distance from PQ 1 for a semicircle

Page 3 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 © University of Cambridge International Examinations 2005 ii) correct line drawn ± 1 mm, ± 1o correct arcs, radius > 4 cm B1 B1 iii) correct area shaded 2 SC1 for shading on left hand side of their ‘mediator’ or inside lines drawn for their b) i) 11 5 a) i) kite 1 ii) correct line BD drawn 1 Allow broken line, one line only iii) 70 2 M1 for 2 80 140 360 − − o.e. b) (p =) 90 (q =) 50 (r =) 50 1 1 1√ f.t. from their q, not strict f.t. c) 128.6 c.a.o. 4 M2 for 180 - 7 360 or 7 180 5× o.e. (may be implied by art 129) +A1 for 128.57 11 6 a) 3 0 0 1,1,1 b) 7 correct points plotted P3√ P2√ for 5 or 6 points ± ½ sm. sq. P1√ for 4 points. not strict f.t. smooth curve through all correct points C1 incorrectly plotted points should be ignored for C1. Minimum curved, not pointed c) -0.8 to -0.7 c.a.o. 1 ignore any y values 2.7 to 2.8 c.a.o. 1

Page 4 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 © University of Cambridge International Examinations 2005 d) 4 0 1,1 e) correct line drawn through (-4,8) and (4,0) 1 complete line f) -1.7 to -1.4 c.a.o. 2.4 to 2.7 c.a.o. 1 1 ignore any y values 14 7 a) i) 16 1 ii) 3x + 8 o.e. 2 M1 for 3x. allow n instead of x. deduct 1 for ‘= x’ or ‘= 0’ or = any number, but allow a different letter b) -9a 1 +5b 1 c) 3a(2 – 3a) 2 M1 for any correct partial factorisation d) a u - v o.e. 2 M1 for v – u seen e) (x=) 2.5 2 M1 for correct multiplication of LHS of one or both equations to equalise coefficients or for a recognisable attempt to eliminate one variable (y=) -3.5 2 M1 for correct substitution of their other value or M2 correct matrix method 13 8 a) i) 22 1 ii) 77 or 2 87 67 + 2 M1 for evidence of ranking seen anywhere. e.g. 67,87 iii) 89 2 M1 for their 12 ∑x

Page 5 Mark Scheme Syllabus Paper IGCSE EXAMINATIONS – NOVEMBER 2004 0580/0581 3 © University of Cambridge International Examinations 2005 b) i) 72 ± 1 80 ± 1 94 ± 1 1 1 1 ii) 1080 ± 5 1200 ± 5 1410 ± 5 1√ 1√ 1√ strict f.t.s for their angle x 15 ± 5 iii) appropriate observation 1 12 9 a) i) 27 to 36 entered correctly 1 ii) a) square 1 b) 100 1 c) n2 c.a.o. 1 allow n x n iii) a) 43 c.a.o. 1 b) 871 2 M1 for 900 – 30 + 1 o.e. b) i) 100 1 ii) 10n c.a.o. 1 allow 10 x n iii) 91 1 vi) 10n – 9 o.e. 1 11 Total 104