Cambridge IGCSE Mathematics 0580 — 2012 May/June Paper 4 · Variant 2

1 For Examiner's Mathematics mark 30 50 35 25 5 39 48 40 10 15 Use English mark 26 39 35 28 9 37 45 33 16 12 The table shows the test marks in Mathematics and English for 10 students. (a) (i) On the grid, complete the scatter diagram to show the Mathematics and English marks for the 10 students. The first four points have been plotted for you. 50 40 30 English mark 20 10 0 5 10 15 20 25 30 35 40 45 50 Mathematics mark [2] (ii) What type of correlation does your scatter diagram show? Answer(a)(ii) [1] (iii) Draw a line of best fit on the grid. [1] (iv) Ann missed the English test but scored 22 marks in the Mathematics test. Use your line of best fit to estimate a possible English mark for Ann. Answer(a)(iv) [1] (b) Show that the mean English mark for the 10 students is 28. Answer(b) [2] (c) Two new students do the English test. They both score the same mark. The mean English mark for the 12 students is 31. Calculate the English mark for the new students. Answer(c) [3]

Mark scheme: Qu. Answers Mark Part Marks 1 (a) (i) 6 correct plots 2 P1 for 4 or 5 correct plots. (ii) Positive 1 (iii) Line of best fit 1 Ruled line at least from x = 5 to x = 48, with at least 3 points on each side and cuts axes between (5, 0) and (0, 20) (iv) English (integer) value on line at 1ft Strict ft from their single ruled line 5 ø x ø 48. M = 22 (b) (26 + 39 + 35 + 28 + 9 + 37 + 45 + 33 M2 M1 for 26 + 39 + 35 + 28 + 9 + 37 + 45 + 33 + 16 + 16 + 12) ÷ 10 + 12, condone one slip or SC1, for at least 2 values eg (26 + 39 +…) ÷ 10 (c) 46 cao www 3 3 M2 for (31 × 12 – 28 × 10) ÷ 2 soi by 92 ÷ 2 or M1 for 31 × 12 soi by 372 or 92

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2 (a) In a sale, Jen buys a laptop for $351.55. For This price is 21% less than the price before the sale. Examiner's Use Calculate the price before the sale. Answer(a) $ [3] (b) Alex invests $4000 at a rate of 8% per year simple interest for 2 years. Bob invests $4000 at a rate of 7.5% per year compound interest for 2 years. Who receives more interest and by how much? Answer(b) receives $ more interest. [6]

Mark scheme: 2 (a) 445 final answer www 3 3 M2 for 351.55 ÷ (1 – 0.21) oe or M1 for 351.55 = (100 – 21) (%) (b) 640 or 4640 2 M1 for 4000 × 0.08 × 2 oe 4622.5 or 622.5 2 M1 for 4000 × (1.075) 2 oe or 4000 × 0.075 (= 300) and (4000 + their 300) × 0.075 and total interest = the sum of their 2 interests. Alex by 17.5(0) cao final answer 2 M1 for S I amount – C I amount or reverse www 6 or simple interest – compound interest or reverse IGCSE – May/June 2012 0580 42

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3 Pablo plants x lemon trees and y orange trees. For Examiner's (a) (i) He plants at least 4 lemon trees. Use Write down an inequality in x to show this information. Answer(a)(i) [1] (ii) Pablo plants at least 9 orange trees. Write down an inequality in y to show this information. Answer(a)(ii) [1] (iii) The greatest possible number of trees he can plant is 20. Write down an inequality in x and y to show this information. Answer(a)(iii) [1] (b) Lemon trees cost $5 each and orange trees cost $10 each. The maximum Pablo can spend is $170. Write down an inequality in x and y and show that it simplifies to x + 2y Y 34. Answer (b) [1] (c) (i) On the grid opposite, draw four lines to show the four inequalities and shade the unwanted region. y For Examiner's 24 Use 22 20 18 16 14 12 10 8 6 4 2 x 0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 [7] (ii) Calculate the smallest cost when Pablo buys a total of 20 trees. Answer(c)(ii) $ [2]

Mark scheme: 3 (a) (i) x > 4 1 (ii) y > 9 1 (iii) x + y < 20 1 (b) 5x + 10y < 170 seen 1 (c) (i) x = 4 ruled 1 Each line long enough to enclose their region y = 9 ruled 1 Condone good freehand or dotted y = 9 must be between 8.8 and 9.2 x + y = 20 ruled 2 B1 for gradient = – 1 or y intercept = 20 or x intercept = 20. Exclude lines parallel to either axis. x + 2y = 34 ruled 2 B1 for y intercept = 17 or x intercept = 34. Exclude lines parallel to either axis. Correct region indicated cao 1 Dependent on all 6 marks for the 4 lines. (ii) 145 cao (from 11, 9) www 2 2 M1 for using 5x + 10y when x + y = 20 and integers (x, y) is in their region IGCSE – May/June 2012 0580 42

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4 (a) For B Examiner's Use A C NOT TO 42° SCALE D O E F A, B, C, D, E and F are points on the circumference of a circle centre O . AE is a diameter of the circle. BC is parallel to AE and angle CAE = 42°. Giving a reason for each answer, find (i) angle BCA, Answer(a)(i) Angle BCA = Reason [2] (ii) angle ACE, Answer(a)(ii) Angle ACE = Reason [2] (iii) angle CFE, Answer(a)(iii) Angle CFE = Reason [2] (iv) angle CDE. Answer(a)(iv) Angle CDE = Reason [2] (b) For Examiner's P Use NOT TO 5 cm SCALE O 12 cm Q In the diagram, O is the centre of the circle and PQ is a tangent to the circle at P. OP = 5 cm and OQ = 12 cm. Calculate PQ. Answer(b) PQ = cm [3] (c) C B D E NOT TO SCALE A G F In the diagram, ABCD and DEFG are squares. (i) In the triangles CDG and ADE, explain with a reason which sides and/or angles are equal. Answer (c)(i) [3] (ii) Complete the following statement. Triangle CDG is to triangle ADE. [1]

Mark scheme: 4 In all parts of (a) candidates may refer to angles marked in diagram. Allow if clear even if reason is more complicated as long as it is full. Reasons dependent on correct answers (a) (i) 42 1 Not alternate segment Alternate oe 1 (ii) 90 1 Allow diameter semicircle oe 1 (iii) 42 1 same arc same segment oe 1 (iv) 138 1 key words must not be spoiled cyclic quad oe 1 (b) 10.9 (10.90 to 10.91) www 3 3 M2 for 12 2 − 5 2 oe i.e explicit or M1 for 12 2 = 5 2 + PQ 2 oe i.e implicit Allow full marks for 119 as final answer Use of trig method must be complete to explicit expression for possible M2 (c) (i) AD = CD and DE = DG 1 Extra pair of sides loses this mark. (Angle) CDG = (angle)ADE 1 Extra pair of angles loses this mark (Sides of) square or 90° + angle ADG R1 As in (a), for all 3 marks allow references to oe diagram if completely clear. R mark dep on at least one pair of sides stated or pair of angles stated (ii) Congruent 1 IGCSE – May/June 2012 0580 42

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5 (a) In Portugal, Miguel buys a book about planets. For The book costs €34.95. Examiner's In England the same book costs £27.50. Use The exchange rate is £1 = €1.17. Calculate the difference in pounds (£) between the cost of the book in Portugal and England. Answer(a) £ [2] (b) In the book, the distance between two planets is given as 4.07 × 1012 kilometres. The speed of light is 1.1 × 109 kilometres per hour. Calculate the time taken for light to travel from one of these planets to the other. Give your answer in days and hours. Answer(b) days hours [3] (c) In one of the pictures in the book, a rectangle is drawn. The rectangle has length 9.3 cm and width 5.6 cm, both correct to one decimal place. (i) What is the lower bound for the length? Answer(c)(i) cm [1] (ii) Work out the lower and upper bounds for the area of the rectangle. Answer(c)(ii) Lower bound = cm2 Upper bound = cm2 [2]

Mark scheme: 5 (a) (£) 2.37 or 2.371 to 2.372 www 2 2 M1 for 34.95 ÷ 1.17 implied by 29.87…or 29.9 or SC1 for 2.77 or 2.78 or 2.775 (b) 154 days 4 hours cao 3 M1 for 4.07 × 1012 ÷ (1.1 × 109) implied by figs 37 or 154. (….) A1 for 3700 seen or 3.7 × 103 seen or 154 16 oe or 154 rem 4 (c) (i) 9.25 1 (ii) Lower = 51.3375 final answer 1 After 0 scored Upper = 52.8275 final answer 1 SC1 for answers reversed or 9.35 and 5.65 seen or 51.3375 and 52.8275 seen

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6 (a) For Examiner's 114° Use 2x° NOT TO SCALE x° (x – 10)° Find the value of x. Answer(a) x = [3] (b) (i) Write the four missing terms in the table for sequences A, B, C and D. Term 1 2 3 4 5 n Sequence A – 4 2 5 8 3n – 7 Sequence B 1 4 9 16 25 Sequence C 5 10 15 20 25 Sequence D 6 14 24 36 50 [4] (ii) Which term in sequence D is equal to 500? Answer(b)(ii) [2] x 2 − 16 (c) Simplify . 2 x 2 + 7 x − 4 Answer(c) [4]

Mark scheme: 6 (a) (x =) 64 www 3 3 B2 for x + 2 x + x = 360 − 114 + 10 or better or M1 for x + 2 x + 114 + x − 10 = 360 (b) (i) –1 1 n 2 oe 1 5n oe 1 2 1 n + 5n oe (ii) 20 2 M1 for their n 2 + 5n = 500 or 20 and 25 seen x − 4 (c) Final answer cao www 4 4 B1 for (x – 4)(x + 4) 2 x − 1 B2 for (2x – 1)(x + 4) or SC1 for (2x + a)(x + b) where either a + 2b = 7 or ab = – 4

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For  3 7 (a) P is the point (2, 5) and =   . Examiner's  −2  Use Write down the co-ordinates of Q. Answer(a) ( , ) [1] (b) D C B NOT TO E SCALE c M O A 3a O is the origin and OABC is a parallelogram. M is the midpoint of AB. = c, = 3a and CE = 1 CB. 3 OED is a straight line with OE : ED = 2 : 1 . Find in terms of a and c, in their simplest forms (i) , Answer(b)(i) = [1] (ii) the position vector of M, Answer(b)(ii) [2] (iii) , Answer(b)(iii) = [1] (iv) . Answer(b)(iv) = [2] (c) Write down two facts about the lines CD and OB. Answer (c) [2]

Mark scheme: 7 (a) (5, 3) 1 (b) (i) 3a + c 1 (ii) 3a + 12 c or 12 (6a + c ) 2 M1 for OM oe e.g OA+AM or correct unsimplified answer (iii) a + c 1 3 1 1 (iv) a + c or (3a + c) 2 M1 for – c + 3× their (iii) or a + 1× their (iii) or 2 2 2 2 2 correct unsimplified answer or any correct route e.g. CE + ED Part (c) dependent on simplified (i) and (iv) (c) (CD) parallel (to OB) oe cao 1dep Dep on (i) = k × (iv) 1 CD = OB oe cao 1dep Dep on (i) = 2 × (iv) must be scalars 2 IGCSE – May/June 2012 0580 42

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8 In all parts of this question give your answer as a fraction in its lowest terms. For Examiner's Use 1 . (a) (i) The probability that it will rain today is 3 What is the probability that it will not rain today? Answer(a)(i) [1] 2 (ii) If it rains today, the probability that it will rain tomorrow is . 5 1 If it does not rain today, the probability that it will rain tomorrow is . 6 Complete the tree diagram. Today Tomorrow Rain Rain No rain Rain No rain No rain [2] (b) Find the probability that it will rain on at least one of these two days. Answer(b) [3] (c) Find the probability that it will rain on only one of these two days. Answer(c) [3]

Mark scheme: 8 Throughout question, penalise non-reduced fraction only once; isw any conversion and allow decimals in working and on branches but not final answers if fractions not seen. 2 (a) (i) 1 3 1 2 2 3 1 5 1 2 3 5 (ii) , , , , , 2 B1 for and and or correctly placed 3 3 5 5 6 6 3 3 5 6 correctly placed For method marks in (b) and (c), ft tree with each probability 0< p <1 4 2 5 1 2 1 (b) cao www 3 3 M2 for 1 – × or + × 9 3 6 3 3 6 1 2 1 3 2 1 or × + × + × 3 5 3 5 3 6 1 2 5 M1 for + × 3 3 6 1 2 1 3 2 1 or two of × , × , × added 3 5 3 5 3 6 14 1 3 2 1 4 1 2 (c) cao www 3 3 M2 for × + × or their – × 45 3 5 3 6 9 3 5 1 3 2 1 M1 for one of × or × from a 3 5 3 6 maximum of two products added.

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9 For F Examiner's Use E Scale 1 : 10 000 H G The diagram is a scale drawing of a park EFGH. The scale is 1 : 10 000. A statue is to be placed in the park so that it is • nearer to G than to H • nearer to HG than to FG • more than 550 metres from F. Construct accurately the boundaries of the region R in which the statue can be placed. Leave in all your construction arcs and shade the region R. [7]

Mark scheme: 9 Accurate ruled perp. bisector with 2 B1 for accurate with no/wrong arcs correct intersecting arcs or M1 for correct intersecting arcs Ignore one extra perp. bisector Accurate ruled angle bisector with 2 B1 for accurate with no/wrong arcs correct intersecting arcs or M1 for correct intersecting arcs Ignore one extra angle bisector Compass drawn arc centre F radius 2 M1 for compass drawn arc centre F 5.5 cm long enough to enclose region Accept dotty lines but not freehand for all three Correct region indicated cao 1 IGCSE – May/June 2012 0580 42

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10 (a) Simplify For Examiner's (i) (2x2y3)3, Use Answer(a)(i) [2] _ 1  27  3 (ii)  6  .  x  Answer(a)(ii) [3] (b) Multiply out and simplify. (3x – 2y)(2x + 5y) Answer(b) [3] (c) Make h the subject of (i) V = πr3 + 2πr2h, Answer(c)(i) h = [2] (ii) V = 3h . Answer(c)(ii) h = [2] (d) Write as a single fraction in its simplest form. x 5x 7x + – 2 3 4 Answer(d) [2]

Mark scheme: 10 (a) (i) 8x 6 y 9 final answer 2 B1 for any two of 8, x 6, y 9 in a single term in answer x 2 1 3 −2 1 (ii) oe but not oe final answer 3 B2 for 2 or 3 x or − 2 as answer − 2 3 3 x x 3 x x 6 1 or B1 for oe as answer or seen 27 27 3 x 6 or SC1 for 3 or x 2 or x – 2 seen in answer (b) 6x 2 + 11xy – 10y 2 final answer 3 B2 for 3 of 6x 2 – 4xy + 15xy – 10y 2 (11xy implies 2 terms) or B1 for 2 of 6x 2 – 4xy + 15xy – 10y 2 V − πr 3 V r (c) (i) or − oe but not triple 2 M1 for correct subtraction or correct division by 2πr 2 2 πr 2 2 2 2 πr seen fractions final answer 2 V 2 2 V  V  (ii) final answer 2 B1 for V = 3h or = h or h =   3 3  3  5x 6x 20x − 21x 10x (d) final answer 2 B1 for 2 of , , oe implied by 12 12 12 12 24 ie 2 with common denominator = at least 6 2

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