Cambridge IGCSE Mathematics (with coursework) 0581 — 2006 Oct/Nov Paper 3 · Variant 1

This document consists of 13 printed pages and 3 blank page. IB06 11_0580_03/3RP  UCLES 2006 [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 2006 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 soft 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 of the 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. Give 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 2006 0580/03/N/06 For Examiner's Use 1 (a) 3 2 2 3 3.14 35 10 24 37 45 88 From the list of numbers above choose one that is (i) an irrational number, Answer(a) (i) [1] (ii) the cube root of 27, Answer(a) (ii) [1] (iii) a multiple of 9, Answer(a) (iii) [1] (iv) a prime number, Answer(a) (iv) [1] (v) a factor of 44, Answer(a) (v) [1] (vi) the product of 6 and 4. Answer(a) (vi) [1] (b) The diagram below shows a sequence of patterns made with small triangular tiles. Pattern number 1 2 3 4 (i) Draw the next pattern in the sequence. [1] (ii) Complete the table below. Pattern number 1 2 3 4 5 6 Number of tiles 1 4 9 [2] (iii) How many tiles will be in the 100th pattern? Answer(b) (iii) [1] (iv) How many tiles will be in the nth pattern? Answer(b) (iv) [1] (v) What is the special name given to the numbers in the second row of the table? Answer(b) (v) [1]

3 © UCLES 2006 0580/03/N/06 [Turn Over For Examiner's Use 2 (a) Complete the table for the equation y = − x2 + x + 2. x −3 −2 −1 0 1 2 3 4 y −10 0 2 2 0 [3] (b) On the grid below draw the graph of y = − x2 + x + 2. y x 3 2 1 4 3 2 1 0 2 1 1 2 3 4 5 6 7 8 9 10 3 [4] (c) On the grid, draw the line of symmetry of your graph. [1] (d) Use your graph to find the maximum value of y. Answer(d) y = [1] (e) Draw the line y = 1 on the grid. [1] (f) Write down the two values of x for which − x2 + x + 2 = 1. or x = Answer(f) x = [2]

4 © UCLES 2006 0580/03/N/06 For Examiner's Use 3 (a) (i) Calculate the interior angle of a regular heptagon (seven-sided polygon). Write down all the figures on your calculator display. Answer(a) (i) [2] (ii) Round your answer to part (a)(i) to 1 decimal place. Answer(a) (ii) [1] (b) 80º 95º xº 3yº NOT TO SCALE The diagram shows four angles around a point. (i) Write down an equation in x and y. Answer(b) (i) [1] (ii) Simplify your equation. Answer(b) (ii) [1] (iii) Find y when x = 65. Answer(b) (iii) y = [2]

5 © UCLES 2006 0580/03/N/06 [Turn Over For Examiner's Use (c) (i) aº 2bº 70º B A C NOT TO SCALE Explain why a + 2b = 110 in the triangle above. Answer(c) (i) [1] (ii) NOT TO SCALE bº aº Explain why a + b = 90 in the semi-circle above. Answer(c) (ii) [1] (iii) Solve the equations a + 2b = 110, a + b = 90. Answer(c) (iii) a = b = [2] (iv) Work out the size of angle ABC in the triangle in part (c)(i). Answer(c) (iv) Angle ABC = [1]

6 © UCLES 2006 0580/03/N/06 For Examiner's Use 4 3 2 1 2 3 4 5 6 7 8 4 y x 0 5 4 3 2 1 1 2 3 4 A B C 1 5 (a) Describe fully the single transformation that maps (i) triangle A onto triangle B, Answer(a) (i) [3] (ii) triangle A onto triangle C. Answer(a) (ii) [2] (b) On the grid above draw (i) the translation of A by the vector       −3 2 , [2] (ii) the rotation of B through 180° about the point (−1, −2). [2]

7 © UCLES 2006 0580/03/N/06 [Turn Over For Examiner's Use 5 A B D C 18 cm 10 cm X 55º NOT TO SCALE The diagram shows a rectangular tile ABCD which has a shaded triangle DXB. DC = 18 centimetres, BC = 10 centimetres and angle ADX = 55°. (a) Calculate the area of triangle BDC. Answer(a) cm2 [2] (b) Calculate the length of AX. Answer(b) cm [2] (c) Calculate the shaded area. Answer(c) cm2 [3] (d) Calculate the length of BD. Answer(d) cm [2]

8 © UCLES 2006 0580/03/N/06 For Examiner's Use 6 Part of the wall brick face 20 cm 10 cm NOT TO SCALE (a) A builder estimates the number of bricks in a wall by dividing the area of the wall by the area of the face of a brick. A brick wall is 10 metres long and 1.5 metres high. Each brick is 20 centimetres long and 10 centimetres high. Calculate how many bricks the builder estimates are in the wall. Show all your working. Answer(a) bricks [3] (b) Another wall will need 720 bricks. The builder adds an extra 5% to this number to allow for mistakes. (i) Calculate how many bricks the builder needs to buy. Answer(b) (i) bricks [2] (ii) Bricks are sold in packs of 100 which can not be split. How many packs should the builder buy? Answer(b) (ii) packs [1] (c) The builder mixes sand and cement in the ratio 5:2 to make mortar. He wants 14 buckets of mortar. (i) How many buckets of sand and how many buckets of cement does he need? buckets of sand and buckets of cement. Answer(c) (i) He needs [2] (ii) One bag of cement fills 3.5 buckets. How many bags of cement must the builder buy? Answer(c) (ii) bags [1]

9 © UCLES 2006 0580/03/N/06 [Turn Over For Examiner's Use 7 3 2 1 1 2 3 4 5 6 7 8 y x 0 A B 6 5 4 3 2 1 1 2 3 4 Two straight lines labelled A and B are shown on the grid above. (a) Find the gradient of line A. Answer(a) [2] (b) The equation of line B can be written as y = mx + c. Find the values of m and c. Answer(b) m = c = [2] (c) (i) On the diagram draw the line which is parallel to B and passes through the point (1,−1). [1] (ii) Write down the equation of this line. Answer(c) (ii) [2]

10 © UCLES 2006 0580/03/N/06 For Examiner's Use 8 (a) Naomi records the sizes of the 34 pairs of shoes that her shop sells in one day. 4 10 5 6 4 8 6 4 7 3 9 7 4 7 3 5 4 6 5 10 7 5 5 6 4 7 7 6 6 5 5 3 5 6 (i) Using the list above complete the frequency table. Shoe size 3 4 5 6 7 8 9 10 Frequency [3] (ii) Calculate the mean of these shoe sizes. Answer(a) (ii) [3] (iii) Find the range of these sizes. Answer(a) (iii) [1] (iv) Find the mode of these sizes. Answer(a) (iv) [1] (v) Work out the median shoe size. Answer(a) (v) [2] (vi) Calculate the percentage of all the pairs of shoes that are size 7. Answer(a) (vi) %. [2] (vii) Naomi orders 306 pairs of shoes to sell in her shop. Estimate how many of these pairs of shoes should be size 7. Answer(a) (vii) [2]

11 © UCLES 2006 0580/03/N/06 [Turn Over For Examiner's Use (b) Findlay draws a bar chart to show how many pairs of shoes he has sold in his shop in one week. Shoe size Frequency 3 5 10 15 4 5 6 7 8 9 10 (i) Use the information in the bar chart to complete the frequency table below. Shoe size 3 and 4 5 and 6 7 and 8 9 and 10 Frequency [2] (ii) Which is the modal class in the frequency table? Answer(b) (ii) [1]

12 © UCLES 2006 0580/03/N/06 For Examiner's Use 9 The sketch shows the positions of three islands A, B and C. B is 150 kilometres due West of A. C is 110 kilometres due North of A. C A B 150 km 110 km NOT TO SCALE North (a) Using a scale of 1 centimetre to represent 20 kilometres draw accurately the triangle ABC. A is marked for you. A [3] (b) A boat sets out from B to sail directly to C. (i) Use your protractor to find the three-figure bearing of B from C. Answer(b) (i) [2]

13 © UCLES 2006 0580/03/N/06 For Examiner's Use (ii) Measure BC on your diagram and hence find the distance in kilometres of B from C. Answer(b) (ii) km [2] (iii) The boat sails at 20 knots. [1 knot is 1.85 kilometres per hour.] How long will the boat take for the first 100 kilometres of the journey? Give your answer in hours and minutes, to the nearest minute. hours Answer(b) (iii) min [4] (iv) The boat takes 45 minutes for the next 18 kilometres. Calculate this speed in kilometres per hour. Answer(b) (iv) km/h [2] (v) A radio beacon at A has a range of 100 kilometres. On your diagram in part (a) draw accurately the locus of points that are 100 kilometres from A. [2] (vi) For how many kilometres is the boat within range of the beacon? Answer(b) (vi) km [2]

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/N/06 BLANK PAGE

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the October/November 2006 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 instructed to award marks. It does not indicate the details of the discussions that took place at an Examiners’ meeting before marking began. 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. The grade thresholds for various grades are published in the report on the examination for most IGCSE, GCE Advanced Level and Advanced Subsidiary Level syllabuses. • CIE will not enter into discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the October/November 2006 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 Syllabus Paper IGCSE - OCT/NOV 2006 0580, 0581 3 © UCLES 2006 Qu. Answer Marks Comments Total 1 (a) (i) √35 1 (ii) 3 1 (iii) 45 1 (iv) 2 or 3 or 37 1 accept any combination (v) 2 1 (vi) 24 1 (b) (i) Correct arrangement of triangles drawn. 1 accept if only 1 internal line missing (ii) 16 25 36 2 1 mark for 2 correct (iii) 10000 or 1 x 104 1 Not 1002 (iv) n2 or n × n 1 accept t = n2 etc. do not accept x2 (v) Square (numbers) 1 accept squares, squared 12 2 (a) –4 –4 –10 3 1 for each correct entry (b) 8 correctly plotted points, within 2 1 square. Smooth curve through 8 points P3ft C1 P2 for 6 or 7 correct. ft P1 for 4 or 5 correct. ft Allow small errors in the points provided shape is maintained. (c) x = 0.5 drawn. 1 must be from (0.5, –9) to curve at least (d) 2.2 to 2.4 1ft (e) y = 1 drawn. 1 must touch curve as min. length (f) (x =) –0.7 to –0.5 (x =) 1.5 to 1.7 1 1 12 3 (a) (i) 128.571…… or 128° 4 3 ′ (….) 2 M1 for 180 – 360/7 oe (ii) 128.6 1 ft Follow through their (a)(i). (b) (i) x + 3y + 80 + 95 = 360 (or better) 1 (ii) x + 3y = 185 oe 1 Both marks may be gained in (b)(i) (iii) 40 2 ft M1 for x correctly substituted into the linear equation. Follow through their (b)(ii) provided linear in x and y. (c) (i) 180° or angle sum of triangle mentioned 1 (ii) Angle in a semi-circle mentioned. 1 (iii) (a =) 70 (b =) 20 1 1 SC1 for a = 20 b = 70 (iv) 40 1ft 2 × their value for b provided 0 < b < 55. 12 4 (a) (i) Enlargement (Scale Factor) 3 (Centre) (2, 4) B1 B1 B1 . (ii) Reflection (in the line) x = 4 B1 B1 (b) (i) Correct translation drawn 2 SC1 for translation by the vector.       −             −      − 3 2 5 . 1 1 2 3 k k (ii) Correct rotation drawn 2 SC1 for any 180° rotation. SC1 for 90° or 270° rotation about (–1, –2) 9

Page 3 Mark Scheme Syllabus Paper IGCSE - OCT/NOV 2006 0580, 0581 3 © UCLES 2006 5 (a) 90 2 M1 for 0.5 × 18 × 10 (b) 14.3 art 2 M1 for 10 × tan 55oe (c) 18.5 to 18.6 3 M1 for 0.5 × 10 × their (b) or M1 18 – their (b) M1 2 1 x 10 x their BX M1 for Their (a) – (0.5 × 10 × their (b)) (d) 20.6 art 2 M1 for √( 182 + 102) oe 9 6 (a) 750cao 3 M1 Figs 10 ÷ figs 20 and figs 15 ÷ figs 10. OR M1 Figs 10 x Figs 15 and Figs 20 x Figs 10 M1 dep bricks in length × bricks in height. M1 dep. area of wall ÷ area of brick. If MO then SC1 for Figs 75 (b) (i) 756 2 M1 for 720 × 1.05 oe (ii) 8 1ft Their (b)(i) rounded up to the number of hundreds (c) (i) 10 4 1 1 (ii) 2 1ft Their cement buckets ÷ 3.5 and rounded up to next whole number 9 7 (a) –1 2 SC1 for 1 SC1 for K k − (b) (m =) 2 (c =) 3 1 1 (c) (i) Correct line drawn. 1 must cross both axes and line A (ii) y = 2x – 3 oe 2ft SC1 for m = 2 or c = –3. Follow through their line for 2 and SC1. 7 8 (a) (i) 3 6 8 7 6 1 1 2 3 2 for 6 or 7 correct –1 if tally marks 1 for 4 or 5 correct (ii) 5.71 art 3 M1 for evidence of size x frequency calculated for the sizes. M1dep for sum of at least 5 ÷ 34 (iii) 7 cao 1 (iv) 5 cao 1 (v) 5.5 2 M1 for evidence of finding the middle shoe size. (Not just an answer of 5 or 6) (vi) 17.6 art 2ft M1 for their 6 ÷ 34 × 100 or 17.65 (vii) 54 or 53 2ft M1 for their 6 ÷ 34 × 306 or ‘53.8….’. or 53.9 (b) (i) 12 25 19 2 2 1 mark for 2 or 3 correct or all correct but not added (ii) 5 and 6 1ft Their class with the highest frequency. –1 for tally marks 17

Page 4 Mark Scheme Syllabus Paper IGCSE - OCT/NOV 2006 0580, 0581 3 © UCLES 2006 9 (a) Correct accurate drawing. (lengths ± 0.2 cm, angles ± 1°) 3 M1 for angle = 90° = BAC. M1 for AB = 7.5cm and AC = 5.5 cm. A1 for completed triangle. (Dependent on at least one M) (b) (i) 233° to 235° 2ft From their diagram. M1 for their angle BCA measured correctly (± 1°) (ii) 182 to 190 2ft Their BC × 20. M1 for their BC (correct is 9.1 cm to 9.5 cm) (iii) 2 (hours) 42 (mins) 4 SC3 for 2.7(0….) M1 for 20 × 1.85 M1 for 100 ÷ their 37 SC2 for 2 hr 7 mins with no method. B1 for their time correctly changed to hours and minutes. (iv) 24 2 M1 for 18 ÷ 0.75 oe (v) Correct circle drawn 2 M1 for partial circle (crossing AB and AC) (vi) 84 to 100 2ft M1 for 4.2 to 5.0 Follow through their diagram, dependent on intersections seen on BC 17 Total marks 104