Cambridge IGCSE Mathematics 0580 — 2007 Oct/Nov Paper 3 · Variant 1

1 Margarita keeps a record of all her marks for science experiments, as shown in the table below. For Examiner's Mark 5 6 7 8 9 10 Use Frequency 1 5 10 9 7 3 (a) (i) How many science experiments did Margarita do? Answer(a)(i) [1] (ii) Write down the mode. Answer(a)(ii) [1] (iii) Find the median. Answer(a)(iii) [1] (iv) Calculate the mean. Answer(a)(iv) [3] (b) Margarita draws a pie chart to show this information. The sectors for her marks of 5, 6, 7 and 8 have already been drawn. 5 6 7 8 (i) Calculate the angle of the sector for her mark of 9. Answer(b)(i) [2] (ii) Complete the pie chart accurately. [1]

Mark scheme: 1 (a) (i) 35 B1 cao (ii) 7 B1 cao (iii) 8 B1 cao (iv) 7.71 art B3 ft M1 for 1x5 + 5x6 + 10x7 + 9x8 + 7x9 + 3x10 attempted M1 for ÷ 35 (ft from (a)(i) but not for 6) SC2 for 7.7 (b) (i) 72 2 M1 for 7/35 x 360 (ft but not for 6) oe (ii) line drawn B1 final line (ft) drawn accurately, 1° accuracy [9]

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2 For y Examiner's Use 7 6 5 4 3 2 T 1 x –7 –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 7 –1 –2 –3 –4 –5 –6 –7  _6  (a) Draw the image of triangle T after translation by the vector   . Label it A. [2]   3  (b) Draw the image of triangle T after reflection in the line y = −1. Label it B. [2] (c) Draw the image of triangle T after rotation through 180° about the point (0, 0). Label it C. [2] (d) Draw the image of triangle T after enlargement, centre (0, 0), scale factor 2. Label it D. [2] (e) Describe clearly the single transformation which maps triangle D onto triangle T. Answer(e) [3]

Mark scheme: 2 all within 1 mm (a) translation B2 (–5,4), (–3,4), (–4,5) drawn SC1 for any other translation not parallel to a axis (b) reflection B2 (1,–3), (3,–3), (2,–4) drawn SC1 for reflection in x=–1 or any y=k (c) rotation B2 (–1,–1), (–3,–1), (–2,–2) drawn SC1 for any 180 rotation or +90, –90 about (0,0) (d) enlargement B2 (2,2), (6,2), (4,4) drawn SC1 for any other enlargement sf=2 or centre (0,0) (e) enlargement B1 (sf=) 1/2 B1 (centre) (0,0) B1 accept O [11] IGCSE – October/November 2007 0580 and 0581 3

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36 , (x ≠ 0). For3 (a) Complete the table for the function y = x Examiner's Use x −6 −5 −4 −3 −2 −1 1 2 3 4 5 6 y −7.2 −9 −18 18 9 7.2 [3] 36 (b) On the grid below, draw the graph of y = for −6 x −1 and 1 x 6. x y 40 30 20 10 x –6 –5 –4 –3 –2 –1 0 1 2 3 4 5 6 –10 –20 –30 –40 [4] (c) Use your graph to find x when y = 21. Answer(c) x = [1] (d) Complete the table for the function y = x2. For Examiner's Use x −6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 y 25 16 4 1 1 4 16 25 [2] (e) On the same grid, draw the graph of y = x2 for −6 x 6. [4] 36(f) Write down the co-ordinates of the point of intersection of the graphs of y = and y = x2. x Answer(f)( , ) [1]

Mark scheme: 3 (a) –6, –12, –36, 36, 12, 6 B3 B1 for ± 36, B1 for ± 12, B1 for ± 6 SC1 for any 3 correct (b) 12 points plotted P3 correct points ft within 1 mm P2 for 10 or 11, P1 for 8 or 9, P1 for 1 correct branch 2 curves drawn C1 must be smooth branches of rectangular hyperbola (c) 1.6 to 1.8 B1 ft (d) 36, 9, 0, 9, 36 B2 B1 for 4 correct (e) 13 points plotted P3 correct points ft within 1 mm P2 for 11 or 12 P1 for 9 or 10 curve drawn C1 must be smooth parabola (f) 3.3, 10.9 B1ft x from 3.2 to 3.4, y from 10.0 to 12.0 [15]

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4 r 2r 5πr 2 The area of the shape is given by the formula A = . 2 (a) Calculate the area when r = 3 cm. Answer(a) A = cm2 [2] (b) Calculate the value of r when A = 200 cm2. Answer(b) r = cm [3] (c) Make r the subject of the formula. Answer(c) [3]

Mark scheme: 4 (a) 70.7 art B2 M1 for 5 x π x 3² / 2 or better (b) 5.05 art B3 M1 for 200 = 5 x π x r² / 2 oe M1 for (r² =) 400 / 5π oe (c) (r =) √2A/5π B3 M1 for any correct x or ÷ of 1 term 2A = 5πr² MA1 for r² = 2A / 5π M1 for square root at end [8]

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2 5 (a) –4 –16 0.12 7 144 7 2 For 3 Examiner's Use From this list of numbers, write down (i) the smallest number, Answer(a)(i) [1] (ii) a natural number, Answer(a)(ii) [1] (iii) a square number, Answer(a)(iii) [1] (iv) an irrational number. Answer(a)(iv) [1] (b) Write down 40 as a product of prime numbers. (1 is not a prime number.) Answer(b) 40 = [2] (c) Three pairs of prime numbers have a sum of 40. One pair is 3 and 37. Find the other two pairs. Answer(c) and and [2]

Mark scheme: 5 (a) (i) –16 B1 cao (ii) 7 or 144 or both B1 (iii) 144 B1 cao (iv) √7 B1 cao (b) 2 x 2 x 2 x 5 B2 B1 for 8x5, 2x20, 4x10, 2x4x5, or list 2, 2, 2, 5 (c) 11, 29 B1 cao 17, 23 B1 cao [8] IGCSE – October/November 2007 0580 and 0581 3

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6 (a) Pencils cost 5 cents each and erasers cost 4 cents each. For Examiner's (i) Work out the total cost of 10 pencils and 7 erasers. Use Answer(a)(i) cents [1] (ii) Write down, in terms of p and e, the total cost of p pencils and e erasers. Answer(a)(ii) cents [1] (b) The cost of a pen is x cents and the cost of a ruler is y cents. 2 pens and 3 rulers have a total cost of 57 cents. 5 pens and 1 ruler have a total cost of 58 cents. (i) Write down two equations in x and y. Answer(b)(i) [2] (ii) Find the value of x and the value of y. Answer(b)(ii) x = y = [4]

Mark scheme: 6 (a) (i) 78 B1 cao (ii) 5p + 4e B1 cao (b) (i) 2x + 3y = 57 B1 5x + y = 58 B1 SC1 for different variables (ii) 15x + 3y = 174 M1 oe, for useful mult. or substitution (2 terms correct) x = 9 A1 cao 18 + 3y = 57 M1 oe, for using first answer correctly and sensibly y = 13 A1 cao [8] www4 ft for M marks only for linear equations in 2 variables √ √

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7 For NOT TO Examiner's SCALE Use A A 3 cm 3 cm 3 cm 3 cm 8 cm B D C B 3 cm C 3 cm Diagram 1 Diagram 2 A physics teacher uses a set of identical triangular glass prisms in a lesson. Diagram 1 shows one of the prisms. Diagram 2 shows the cross-section of one prism. The triangle ABC is equilateral, with sides of length 3 cm and height AD. (a) (i) Calculate the length of AD. Answer(a)(i) cm [2] (ii) Calculate the area of triangle ABC. Answer(a)(ii) cm2 [2] (iii) The length of the prism is 8 cm. Calculate the volume of the prism. Answer(a)(iii) cm3 [2] (b) After the lesson, the glass prisms are put into a box, which is also a triangular prism. For The cross-section is an equilateral triangle, with sides of length 9 cm. Examiner's The length of the box is 16 cm. Use NOT TO SCALE 9 cm 9 cm 16 cm 9 cm (i) Work out the largest number of glass prisms that can fit into the box. Answer(b)(i) [2] (ii) Sketch a net of the box. (Accurate construction is not required.) [1] (iii) Calculate the surface area of the box. Answer(b)(iii) cm2 [6] (iv) The box was made out of plastic, which cost 6 cents per square centimetre. To make the box, 540 cm2 of plastic was bought. Calculate the total cost of the plastic, giving your answer in dollars. Answer(b)(iv) $ [2]

Mark scheme: 7 (a) (i) 2.60 art or 2.6 B2 M1 for √(3²–1.5²) or better (√6.75) oe (ii) 3.90 art or 3.9 B2 ft M1 for 0.5 x 3 x their(a)(i) (iii) 31.2 art B2 ft M1 for 8 x their (a)(ii) (b) (i) 18 www2 M1 for 9 triangles implied, or 2 x k, or attempted sketch (ii) reasonable sketch B1 shows 3 rectangles, 2 triangles in reasonable proportion (iii) area of "rectangle" M1 for 16 x 9, 144, 3 x 9 x 16, 27 x 16, 432 height of triangle M1 for √(9²–4.5²), √60.75, 7.79, 7.8, 3 x (a)(i) ft or trig area of triangle M1 for 0.5 x height (ft but not 9) x 9, 35.1, 70.2, 70.1 OR M2 for 9 x 3.90, 9 x their (a)(ii), 35.1 , 70.2, 70.1 total area M1 3 rectangles and 2 triangles, 432 + 70.2 or 70.1 soi 502 art A2 if M<3 then add SC3 for 502 art with no wrong working seen (iv) 32.4(0) B2 M1 for 540 x 6 or figs 324 [17]

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8 Carlos is in a class of 12 students. For He compares the results of the students in a mathematics test with their results in a history test. Examiner's The table shows these results. Use Student A B C D E F G H I J K L Mathematics mark 17 8 11 15 14 19 9 12 19 18 13 15 History mark 10 13 10 8 11 7 14 11 10 11 11 10 (a) A student is chosen at random. What is the probability that the student scored more than 10 marks (i) in mathematics, Answer(a)(i) [1] (ii) in mathematics and in history, Answer(a)(ii) [1] (iii) in at least one subject? Answer(a)(iii) [1] (b) The mean mathematics mark is 14.2. Calculate the mean history mark. Answer(b) [2] (c) 15 14 13 12 11 History 10 mark 9 8 7 0 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Mathematics mark (i) On the grid, plot the points to show the results of the 12 students. [3] (ii) Draw a line of best fit. [1] (iii) What type of correlation does this show? Answer(c)(iii) [1]

Mark scheme: 8 (a) (i) 10 / 12. B1 oe 2 sf for decimals and %'s (with sign) throughout (ii) 4 / 12. B1 oe (iii) 12 / 12. B1 oe (b) 10.5 B2 M1 for (10+13+10+8+ ) / 12 or 126 / 12 (c) (i) 12 points plotted B3 B2 for 11, B1 for 10 (ii) ruled line B1 reasonable, at least from 8 to 19 (iii) negative B1 cao [10] IGCSE – October/November 2007 0580 and 0581 3

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9 For Examiner's Use Q T P The scale drawing shows a map of a town. The positions of the town hall, T, and two post offices, P and Q, are marked. On the scale drawing, 1 centimetre represents 200 metres. (a) A new post office in the town is to be built so that it is 800 m from T and equidistant from P and from Q. (i) On the scale drawing, draw the locus of points which are 800 m from T. [1] (ii) On the scale drawing, using a straight edge and compasses only, construct the locus of points which are equidistant from P and from Q. [2] (iii) Label the position of the new post office R. [1] (iv) Find the actual distance between post offices P and R. Answer(a)(iv) m [2] (b) On the scale drawing, draw straight lines to make triangle PQT. Using a straight edge and compasses only, construct the locus of points which are equidistant from PT and from QT. [2] (c) On the scale drawing, shade the region inside triangle PQT, where points are nearer to Q than to P and nearer to PT than to QT. [2] Question 10 is printed on the next page.

Mark scheme: 9 (a) (i) arc B1 full arc, centre T, radius 4 cm, must cover whole of town (ii) locus B2 must be accurate perpendicular bisector of PQ must show 2 pairs of arcs SC1 for accurate without arcs or with 2 arcs just oor (iii) R labelled B1 ft if possible (iv) 640 to 700 m B2 ft SC1 for 3.2 to 3.5 cm (ft) (b) locus B2 must be accurate bisector of angle T must show all arcs SC1 for accurate without arcs or with all arcs just oor (c) correct shading B2 must be a quadrilateral dependent on at least SC1 in (a)(ii) and (b) [10]

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10 For Examiner's Use Diagram 1 Diagram 2 Diagram 3 Diagram 4 Diagram 5 Look at the sequence of five diagrams above. Diagram 1 has 2 dots and 1 line. Diagram 2 has 6 dots and 7 lines. The numbers of dots and lines in each of the diagrams are shown in the table below. Diagram number 1 2 3 4 5 6 7 Number of dots 2 6 12 20 30 Number of lines 1 7 17 31 49 (a) Fill in the empty spaces in the table for Diagrams 6 and 7. [4] (b) How many dots are there in Diagram n? Answer(b) [2] (c) The number of lines in Diagram n is 2n2 – 1. Which diagram has 287 lines? Answer(c) [2]

Mark scheme: 10 (a) 42, 56 B1B1 cao 71, 97 B1B1 cao (b) n (n + 1) oe B2 M1 for attempt at length x width involving n or n'th (n'th + 1) or k (k + 1) where k is any variable (c) 12 B2 M1 for 2 n² – 1 = 287 [8]

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