Cambridge IGCSE Mathematics (with coursework) 0581 — 2007 May/June Paper 3 · Variant 1

This document consists of 12 printed pages. IB07 06_0580_03/5RP © UCLES 2007 [Turn over *9910953725* For Examiner's Use UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MATHEMATICS 0580/03, 0581/03 Paper 3 (Core) May/June 2007 2 hours Candidates answer on the Question Paper. Additional Materials: Electronic calculator Mathematical tables (optional) Geometrical instruments Tracing paper (optional) 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 fluid. 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 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. 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 104. www.XtremePapers.com

2 © UCLES 2007 0580/03/J/07 For Examiner's Use 1 (a) Find the value of (i) 50, Answer(a)(i) [1] (ii) the square root of 64, Answer(a)(ii) [1] (iii) the cube root of 64, Answer(a)(iii) [1] (iv) the integer closest in value to (1.8)3. Answer(a)(iv) [1] (b) Write down (i) a common factor of 15 and 27, which is greater than 1, Answer(b)(i) [1] (ii) a common multiple of 10 and 12. Answer(b)(ii) [1] (c) (i) Two of the factors of 2007 are square numbers. One of these is 1. Find the other square number. Answer(c)(i) [1] (ii) Write down the two factors of 2007 which are prime. Answer(c)(ii) and [2]

3 © UCLES 2007 0580/03/J/07 [Turn over For Examiner's Use 2 Marguerite earns $336 per month. She divides her earnings between bills, food, savings and personal spending. (a) Her bills take 2 7 of her earnings. Show that $240 is left for her other items. Answer(a) [2] (b) She divides the $240 between food, savings and personal spending in the ratio 5 : 3 : 4. Calculate how much she spends on food. Answer(b) $ [2] (c) She saves the same amount each month. Show that she saves $720 in one year. Answer(c) [2] (d) She invests the $720 in a bank which pays 6% per year compound interest. How much will this be worth after 2 years? Answer(d) $ [3]

4 © UCLES 2007 0580/03/J/07 For Examiner's Use 3 (a) Kinetic energy, E, is related to mass, m, and velocity, v, by the formula E = 1 2 mv2. (i) Calculate E when m = 5 and v = 12. Answer(a)(i) E= [2] (ii) Calculate v when m = 8 and E = 225. Answer(a)(ii) v = [2] (iii) Make m the subject of the formula. Answer(a)(iii) m = [2] (b) Factorise completely xy2 – x2y. Answer(b) [2] (c) Solve the equation 3(x – 5) + 2(14 – 3x) = 7. Answer(c) x = [3] (d) Solve the simultaneous equations 4x + y = 13, 2x + 3y = 9. Answer(d) x = y = [3]

5 © UCLES 2007 0580/03/J/07 [Turn over For Examiner's Use 4 (a) The table shows corresponding values of x and y for the function y = x 60 (x ≠ 0). x −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 y −12 −15 −30 60 12 10 [2] (i) Fill in the missing values of y in the table above. (ii) Plot the points on the grid below and draw the graph for −6 x −1 and 1 x 6. y x 60 50 40 30 20 10 –10 –20 –30 –40 –50 –60 –6 –5 –4 –3 –2 –1 1 0 2 3 4 5 6 [4] (b) Write down the order of rotational symmetry of the graph. Answer(b) [1] (c) Draw the lines of symmetry of the graph on the grid. [2] (d) One line of symmetry intersects the graph at two points. (i) Write down the co-ordinates of these two points. Answer(d)(i) ( , ) and ( , ) [2] (ii) Write down the equation of this line of symmetry. Answer(d)(ii) [1] (e) Find the gradient of the other line of symmetry. Answer(e) [1]

6 © UCLES 2007 0580/03/J/07 For Examiner's Use 5 A bag contains 24 discs. 10 discs are red, 9 discs are green and 5 discs are yellow. (a) The number of discs of each colour can be shown by three sectors on a pie chart. The sector angle for the red discs is 150°. Work out the sector angle for (i) the green discs, Answer(a)(i) [1] (ii) the yellow discs. Answer(a)(ii) [1] (iii) Complete the pie chart below and label the sectors. [2]

7 © UCLES 2007 0580/03/J/07 [Turn over For Examiner's Use (b) A disc is chosen at random. Find, as a fraction, the probability of each of the following events. (i) Event A: the disc is red. Answer(b)(i) [1] (ii) Event B: the disc is red or yellow. Answer(b)(ii) [1] (iii) Event C: the disc is not yellow. Answer(b)(iii) [1] (c) Impossible Probability Scale Certain … … (c)(i) (c)(ii) The diagram shows a horizontal probability scale. Write on the dotted lines in the diagram, the probability of (i) an impossible event, [1] (ii) a certain event. [1] (d) Using the notation, A, B and C, mark the positions of your three answers in part (b) on the Probability Scale diagram in part (c). [3]

8 © UCLES 2007 0580/03/J/07 For Examiner's Use 6 56° D C A B E O NOT TO SCALE 6 m 6 m ABCED is the cross-section of a tunnel. ABCD is a rectangle and DEC is a semi-circle. O is the mid-point of AB. OD = OC = 6 m and angle DOC = 56°. (a) (i) Show that angle COB = 62°. Answer(a)(i) [1] (ii) Calculate the length of OB. Answer(a)(ii) OB = m [2] (iii) Write down the width of the tunnel, AB. Answer(a)(iii) AB= m [1] (iv) Calculate the length of BC. Answer(a)(iv) BC = m [2]

9 © UCLES 2007 0580/03/J/07 [Turn over For Examiner's Use (b) Calculate the area of (i) the rectangle ABCD, Answer(b)(i) m2 [2] (ii) the semi-circle DEC, Answer(b)(ii) m2 [2] (iii) the cross-section of the tunnel. Answer(b)(iii) m2 [1] (c) The tunnel is 500 metres long. (i) Calculate the volume of the tunnel. Answer(c)(i) m3 [2] (ii) A car travels through the tunnel at a constant speed of 60 kilometres per hour. How many seconds does it take to go through the tunnel? Answer(c)(ii) s [3]

10 © UCLES 2007 0580/03/J/07 For Examiner's Use 7 A B C D A B C D O l A quadrilateral ABCD, a line l and a point O are shown on the grid above. (a) Write down the mathematical name for the quadrilateral ABCD. Answer(a) [1] (b) On the grid above, draw the images of the quadrilateral ABCD under the following transformations. (i) Translation by the vector 9 3         − . Label this image P. [2] (ii) Reflection in the line l. Label this image Q. [2] (iii) Rotation, centre A, through 90° anti-clockwise. Label this image R. [2] (iv) Enlargement, centre O and scale factor 3. Label this image S. [3]

11 © UCLES 2007 0580/03/J/07 [Turn over For Examiner's Use 8 O P A B R The diagram shows a circular garden, centre O. A straight path AB touches the circle at P. (a) (i) Draw on the diagram the diameter PQ and label the point Q. [1] (ii) Without measuring, write down the size of angle APQ. Answer(a)(ii) Angle APQ= [1] (iii) The point R is marked on the circumference of the circle. Draw the lines PR and QR. [1] (iv) Write down the reason why the angle PRQ is 90°. Answer(a)(iv) [1] (b) Showing all your construction lines, use a straight edge and compasses only to construct (i) the perpendicular bisector of QR, [2] (ii) the bisector of angle PRQ. [2] (c) Shade the region of the garden between PQ and QR which is closer to R than to Q and closer to RQ than to RP. [2] Question 9 is on the next page.

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 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 2007 0580/03/J/07 For Examiner's Use 9 In the pattern below each diagram shows a letter E formed by joining dots. Diagram 1 Diagram 2 Diagram 3 Diagram 4 (a) Draw the next letter E in the pattern. [1] (b) Complete the table showing the number of dots in each letter E. Diagram 1 2 3 4 5 Dots 8 15 [3] (c) How many dots make up the letter E in (i) Diagram 10, Answer(c)(i) [2] (ii) Diagram n? Answer(c)(ii) [2] (d) The letter E in Diagram n has 113 dots. Write down an equation in n and use it to find the value of n. Answer(d) n = [3]

UNIVERSITY OF CAMBRIDGE INTERNATIONAL EXAMINATIONS International General Certificate of Secondary Education MARK SCHEME for the May/June 2007 question paper 0580 and 0581 MATHEMATICS 0580/03 and 0581/03 Paper 3 (Core), maximum raw mark 104 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. 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 discussions or correspondence in connection with these mark schemes. CIE is publishing the mark schemes for the May/June 2007 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 – May/June 2007 0580/0581 03 © UCLES 2007 1 (a) (i) (ii) (iii) (iv) (b) (i) (ii) (c) (i) (ii) 1 8 or −8 or ±8 4 6 3 Multiple of 60 9 3 and 223 B1 B1 B1 B1 B1 B1 B1 B1,B1 Not −4 [9] 2 (a) 336 336 7 2 × − or 336 7 5 × M1 (=) 240 E1 240 must be seen for this mark (b) 5 ÷ their(5 + 4 + 3) × 240 100 M1 A1cao www 2 (c) 3 ÷ their(5 + 4 + 3) × 240 × 12 M1 Allow 2880 for 240 × 12 and 4 1 for 12 3 . (=) 720 E1 720 must be seen for this mark (d) 720 × 1.06 2 oe M2 Implied by 88.99(2) or 89(total interest)seen M1 for 720 × 1.06 (implied by 763.2 seen) 808.99(2) or 809 A1 SC1 for 806.(4) (Simple Interest) www 3 for 808.99(2) or 809 [9]

Page 3 Mark Scheme Syllabus Paper IGCSE – May/June 2007 0580/0581 03 © UCLES 2007 3 (a) (i) 360 B2 M1 for 2 1 × 5 × 122oe (ii) 7.5oe B2 M1 for 225 /4 oe (implied by 56.25) (iii) 2 2 v E or 2 2 1 v E B2 B1 for 2E or 2 1 E or division by v2 (b) xy( y – x) final answer B2 B1 for x(y2 – xy) or y(xy – x2) SC1 for xy(y + x) (c) 3x – 15 + 28 – 6x (= 7) 13 – 3x (= 7) x= 2 MA1 M1ft A1cao Independent ax + b (=7) from their expansion www 3 (d) Equating coefficients of x or y, or equivalent method. 5y = 5 oe or 10x = 30 oe x = 3, y = 1 M1 A1 A1 or a correctly substituted substitution. E.g. y = 13 – 4x ⇒ 2x + 3(13 – 4x) = 9 www 3 [14] 4 (a) (i) (ii) (b) (c) (d) (i) (ii) (e) −10, −20, −60, 30, 20, 15 Their 12 points plotted correctly. Smooth curves through all points. 2 Correct lines ruled (2.4 to 2.5, 24 to 25) (−2.4 to −2.5, −24 to −25) y = 10x oe −10 B2 P3ft C1 B1 B1,B1 B1ft B1ft B1 B1 B1 for –20 (x = –3) or 20 (x = 3) P2ft for 10 or 11 points correct. P1ft for 8 or 9 points or 1 quadrant correct. Two distinct curves; no part of curves between x = –1 and x = 1 Minimum length from x = –3 to x = 3. ft their points of intersection ft their points of intersection cao cao [13]

Page 4 Mark Scheme Syllabus Paper IGCSE – May/June 2007 0580/0581 03 © UCLES 2007 5 (a) (i) (ii) (iii) 135 (green) 75 (yellow) Ruled lines correct to 2° 3 correctly labelled sectors B1 B1 B1ft B1 Only if (a)(i) + (a)(ii) = 210°. Independent of previous marks (b) (i) 24 10 oe B1 Accept decimals, percentages (ii) 24 15 oe B1 (iii) 24 19 oe B1 (c) (i) 0 B1 SC1 for 12 0 and 12 12 or 24 0 and 24 24 (ii) 1 B1 (d) Labelled arrows correctly positioned by eye B3ft 1 mark for each. ft their probabilities from (b). [12] 6 (a) (i) (ii) (iii) (iv) (b)(i) (ii) (iii) (c)(i) (180 – 56)/2 art 2.82 5.63 to 5.64 5.3 or art 5.30 29.8 to 29.9 art 12.5 42.3 to 42.4 21100 to 21200 B1 B2 B1ft B2 B2ft B2ft B1ft B2ft Alt. 90 − (56 ÷ 2) M1 for 6cos 62° (implied by 2.8) Long method must be complete. 2 × their (a)(ii) M1 for 6sin 62°oe Long method must be complete. M1 for their (a)(iii) × (a)(iv) M1 for 0.5 × π × (their (a)(ii)2) ft is their (b)(i) + (b)(ii) M1 for their (b)(iii) × 500 (ii)       × 1000 3600 60 500 oe M2 M1 for figs 500 ÷ figs 60 30 A1 SC2 for answer of 2 1 min or SC1 for1km per minute seen. www B3 [16]

Page 5 Mark Scheme Syllabus Paper IGCSE – May/June 2007 0580/0581 03 © UCLES 2007 7 (a) Trapezium B1 (b) (i) Translation 9 across, 3 down B2 B1 for 9 across or 3 down or      − 9 3 (ii) Correct reflection B2 B1 any reflection of ABCD in a line parallel to l. (iii) Correct rotation B2 B1 90° clockwise rotation of ABCD about A (iv) Correct enlargement B3 B1 any enlargement of ABCD and B1 any enlargement of ABCD SF 3 or B1 any enlargement of ABCD centre O (not penalise lack of labelling provided intention clear) [10] 8 (a) (i) (ii) (iii) (iv) (b) (i) (ii) (c) Diameter from P through O to Q 90 P to R and Q to R ruled. (angle in a ) semi-circle Bisector of QR with arcs. Bisector of PRQ with arcs. Correct Shading B1 B1cao B1 B1 B2 B2 2 Angle on a diameter. Half the angle at the centre. SC1 if accurate without arcs. Maximum errors 2mm from mid-point and 2° from perpendicular. SC1 if accurate without arcs. Maximum error 2° in line from R. If wrong line and/or angle used treat as misread each time. Dep. on B2 in (b)(i) and (b)(ii). SC1 for ‘correct’ shading but dependent on at least SC1 in (b)(i) and (b)(ii). [10]

Page 6 Mark Scheme Syllabus Paper IGCSE – May/June 2007 0580/0581 03 © UCLES 2007 9 (a) (b) (c) (i) (ii) (d) Letter E correctly drawn 22, 29, 36 71 7n + 1 or 8 + (n – 1) × 7 oe Their (c)(ii) = 113 Full method of solution of their equation. 16 B1 B3 B2 B2 B1ft M1ft A1cao B1 for each correct number. B1 for 7 × 10 + 1 or 8 + 9 × 7 seen. SC1 for 7n + k seen. (k is an integer) oe ft any expression involving n. ft only a linear equation. (113 – k)/ ‘7’ www B2 [11]