Cambridge IGCSE Mathematics 0580 — 2020 May/June Paper 4 · Variant 3

1 (a) Campsite fees (per day) Tent ............. $15.00 Caravan ....... $25.00 The sign shows the fees charged at a campsite. Today there are 54 tents and 18 caravans on the site. Calculate the fees charged today. $ ................................................. [2] (b) In September the total income at the campsite was $37 054. This was a decrease of 4.5% on the total income in August. Calculate the total income in August. $ ................................................. [2] (c) The visitors to the campsite today are in the ratio men : women = 5 : 4 and women : children = 3 : 7. (i) Calculate the ratio men : women : children in its simplest form. ................... : ................... : ................... [2] (ii) Today there are 224 children at the campsite. Calculate the total number of men and women. .................................................. [3] (d) The space allowed for each tent is a rectangle measuring 8 m by 6 m, each correct to the nearest metre. Calculate the upper bound for the area of the space allowed for each tent. ............................................ m2 [2] (e) The value of the campsite has increased exponentially by 1.5% every year since it opened 30 years ago. Calculate the value of the campsite now as a percentage of its value 30 years ago. ............................................. % [2]

Mark scheme: Question Answer Marks Partial Marks 1(a) 1260 2 M1 for 15 × 54 + 25 × 18 1(b) 38 800 2  4.5  M1 for 37054 ÷  1 −  oe  100  1(c)(i) 15 : 12 : 28 2 M1 for correct attempt to find a common multiple for the women oe 1(c)(ii) 216 3 M2 for 224 ÷ their 28 × their (15 + 12) or M1 for 224 ÷ their 28 1(d) 55.25 2 M1 for 8 + 0.5 or 6 + 0.5 seen 1(e) 156 or 156.3… 2 30  1.5  M1 for  1 +   100 

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2 y 10 9 8 7 6 B 5 A 4 3 2 1 – 10 – 9 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 9 10 x – 1 – 2 C – 3 – 4 – 5 – 6 – 7 – 8 – 9 – 10 (a) (i) Draw the image of triangle A after a reflection in the line y =- x . [2] - 2 (ii) Draw the image of triangle A after a translation by the vector [2] e- 9o. (b) Describe fully the single transformation that maps (i) triangle A onto triangle B, ............................................................................................................................................. ............................................................................................................................................. [3] (ii) triangle A onto triangle C. ............................................................................................................................................. ............................................................................................................................................. [3]

Mark scheme: 2(a)(i) triangle with vertices at 2 B1 for correct reflection in y = x (−2, −1) (−8, −1) (−2, −5) 2(a)(ii) triangle with vertices at 2  k   − 2  B1 for translation by   or   (−1, −1) (−1, −7) (3, −7)  − 9   k  2(b)(i) Enlargement 3 B1 for each [centre] (−7, 8) [sf] ½ 2(b)(ii) Rotation 3 B1 for each [centre] (0, 0) 90° clockwise oe

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3 (a) Here is some information about the masses of potatoes in a sack: • The largest potato has a mass of 174 g. • The range is 69 g. • The median is 148 g. • The lower quartile is 121 g. • The interquartile range is 38 g. On the grid below, draw a box-and-whisker plot to show this information. 100 110 120 130 140 150 160 170 180 Mass (g) [4] (b) The table shows the marks scored by some students in a test. Mark 5 6 7 8 9 10 Frequency 8 2 12 2 0 1 Calculate the mean mark. ................................................. [3]

Mark scheme: 3(a) correct diagram 4 B1 for median line correctly drawn at 148 B1 for 105 soi B1 for whisker at 159 soi 3(b) 6.48 3 M1 for (5 × 8) + (6 × 2) + (12 × 7) + … M1dep for their ∑fx ÷ their (8 + 2 + 12 + 2 + 0 + 1)

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4 (a) Solve the inequality. 3m + 12 G 8m - 5 ................................................. [2] (b) Solve the equation. 2x + 5 14 = 3 - x 15 x = ................................................ [3] (c) Solve the simultaneous equations. You must show all your working. y = 4 - x x 2 + 2y 2 = 67 x = .................... , y = .................... x = .................... , y = .................... [6]

Mark scheme: 4(a) m ≥ 3.4 oe final answer 2 M1 for 12 + 5 ≤ 8m – 3m or better or 3m – 8m ≤ –5 – 12 or better 4(b) x = − 0.75 oe 3 M1 for 15 ( 2 x + 5 ) = 14 ( 3 − x ) B1 for 30 x + 75 = 42 − 14 x or better 4(c) 3 x 2 − 16 x − 35[ = 0] or M3 M1 for x 2 + 2 ( 4 − x ) 2 = 67 3 y 2 − 8 y − 51[ = 0] 2 2 or ( 4 − y ) + 2 y = 67 seen B1 for 16 − 8x + x 2 or 16 − 8y + y 2 (3x + 5)(x – 7) [= 0] M1 or for correct factors for their equation or (3y – 17)(y + 3)[= 0] or for correct use of quadratic formula or completing the square for their equation x = 7, y = −3 B2 5 B1 for x = 7, x = − 3 5 2 2 x = − , y = 5 or for y = −3, y = 5 3 3 3 or for a correct pair of x and y values

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5 All the lengths in this question are in centimetres. x + 1 A F D NOT TO 2x E SCALE x + 3 B C 4x – 5 The diagram shows a shape ABCDEF made from two rectangles. The total area of the shape is 342 cm2. (a) Show that x 2 + x - 72 = 0 . [5] (b) Solve by factorisation. x 2 + x - 72 = 0 x = .................... or x = .................... [3] (c) Work out the perimeter of the shape ABCDEF. ............................................ cm [2] (d) Calculate angle DBC. Angle DBC = ................................................ [2]

Mark scheme: 5(a) ( 4 x − 5 )( x + 3 ) + ( x + 1)( x − 3 ) = 342 M2 M1 for ( 4 x − 5 )( x + 3 ) or ( x + 1)( x − 3 ) or or for 2 x ( 4 x − 5 ) or ( 3 x − 6 )( x − 3 ) 2 x ( 4 x − 5 ) − ( 3 x − 6 )( x − 3 ) = 342 4 x 2 + 12 x − 5 x − 15 oe and M2 M1 for each x 2 + x − 3 x − 3 oe seen OR 8 x 2 − 10 x and 3 x 2 − 15 x + 18 seen 5 x 2 + 5 x − 18 = 342 leading to A1 no errors or omission x 2 + x − 72 = 0 5(b) ( x + 9 )( x − 8 ) M2 B1 for (x + a)(x + b) where ab = – 72 or a + b = 1 and a, b are integers 8, −9 B1 5(c) 86 2 FT for 12 × their x − 10 (x positive) B1 for any one of 27, 11, 16 seen or for 2 x + 2 x + 4 x − 5 + 4 x − 5 oe or better soi 5(d) 22.2 or 22.16 to 22.17 2 11 their x + 3 M1 for tan = or 27 4 × their x − 5

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6 (a) C 54° x° NOT TO SCALE 5.3 cm 11 cm G 6.9 cm 42° A B The diagram shows triangle ABC with point G inside. CB = 11 cm, CG = 5.3 cm and BG = 6.9 cm. Angle CAB = 42° and angle ACG = 54°. (i) Calculate the value of x. x = ................................................ [4] (ii) Calculate AC. AC = ........................................... cm [4] (b) NOT TO 2.5 cm SCALE 15 cm Water flows at a speed of 20 cm/s along a rectangular channel into a lake. The width of the channel is 15 cm. The depth of the water is 2.5 cm. Calculate the amount of water that flows from the channel into the lake in 1 hour. Give your answer in litres. ........................................ litres [4]

Mark scheme: 6(a)(i) 29.5 or 29.50… 4 112 + 5.32 − 6.9 2 M2 for 2 × 11 × 5.3 or M1 for 6.92 = 112 + 5.32 − 2 × 11 × 5.3 cos x A1 for 0.87[0…] oe 6(a)(ii) 13.4 or 13.38… 4 B1FT 84 − their (a)(i) 11 M2 for × sin their 54.5 sin42 or M1 for implicit form 6(b) 2700 4 M2 for 15 × 2.5 × 20 × 60 × 60 or M1 for 15 × 2.5 × 20 M1 for their volume ÷ 1000 If 0 scored, SC1 for figs 27 with no working

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7 On any Saturday, the probability that Arun plays football is 3. 4 On any Saturday, the probability that Bob plays football is 2. 5 (a) (i) Complete the tree diagram. Arun Bob ............... Plays Plays ............... Does not play ............... ............... Plays Does not ............... play Does not play ............... [2] (ii) Calculate the probability that, one Saturday, Arun and Bob both play football. ................................................. [2] (iii) Calculate the probability that, one Saturday, either Arun plays football or Bob plays football, but not both. ................................................. [3] (b) Calculate the probability that Bob plays football for 2 of the next 3 Saturdays. ................................................. [3] (c) When Arun plays football, the probability that he scores the winning goal is 1. 7 Calculate the probability that Arun scores the winning goal one Saturday. ................................................. [2]

Mark scheme: 7(a)(i) 3 1 2 3 2 3 2 B1 for one correct pair , , , 4 4 5 5 5 5 7(a)(ii) 3 2 FT their tree diagram oe 3 2 10 M1 for × 4 5 7(a)(iii) 11 3 3 3 1 2 oe M2 for × + × 20 4 5 4 5 3 3 1 2 or M1 for × or × 4 5 4 5 7(b) 36 3  2 2 3 oe M2 for × × 3 oe   125  5  5  2 ×2 3 or M1 for    5  5 7(c) 3 2 3 1 oe M1 for × 28 4 7

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8 (a) The interior angle of a regular polygon with n sides is 150°. Calculate the value of n. n = ................................................ [2] M (b) (i) K, L and M are points on the circle. KS is a tangent to the circle at K. KM is a diameter and NOT TO triangle KLM is isosceles. SCALE Find the value of z. L z° K S z = ................................................ [2] (ii) AT is a tangent to the circle at A. Find the value of x. x° NOT TO SCALE 27° 58° A T x = ................................................ [2] (iii) G y° NOT TO SCALE H F 108° J E F, G, H and J are points on the circle. EFG is a straight line parallel to JH. Find the value of y. y = ................................................ [2] (c) C N NOT TO SCALE D O A B M A, B, C and D are points on the circle, centre O. M is the midpoint of AB and N is the midpoint of CD. OM = ON Explain, giving reasons, why triangle OAB is congruent to triangle OCD. ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... ..................................................................................................................................................... [3]

Mark scheme: 8(a) 12 2 ( n − 2 ) × 180 360 M1 for 150 = or oe n 180 − 150 8(b)(i) 45 2 B1 for angles at M or K = 45 or angle at L = 90 8(b)(ii) 85 2 B1 for either angle in alt segment = 58 8(b)(iii) 72 2 B1 for either angle at J or H=108 or angle at F=72 8(c) OA = OB = OC = OD B1 Radii AB = CD B1 chords equidistant from centre are equal SSS implies congruent B1

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9 (a) The equation of line L is 3x - 8y + 20 = 0 . (i) Find the gradient of line L. ................................................. [2] (ii) Find the coordinates of the point where line L cuts the y-axis. ( ................... , ................... ) [1] (b) The coordinates of P are (-3, 8) and the coordinates of Q are (9, -2). (i) Calculate the length PQ. ................................................. [3] (ii) Find the equation of the line parallel to PQ that passes through the point (6, -1). ................................................. [3] (iii) Find the equation of the perpendicular bisector of PQ. ................................................. [4]

Mark scheme: 9(a)(i) 3 2 M1 for 8 y = 3 x + 20 or better 8 9(a)(ii) (0, 2.5) oe 1 (b)(i) 15.6 or 15.62… 3 2 2 M2 for ( 9 −−3 ) + ( −−2 8 ) oe seen 2 2 or M1 for ( 9 −−3) or ( −−2 8 ) oe seen 9(b)(ii) 5 3 −−2 8 y = − x + 4 oe M1 for gradient oe 6 9 −−3 M1 for substituting (6, −1) into a linear equation oe 9(b)(iii) 6 3 4 5  y = x − oe M1 for gradient −1 / their −   5 5  6  B1 for midpoint at (3, 3) M1 for their midpoint substituted into y = their m × x + c oe

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10 (a) The diagrams show the graphs of two functions. Write down each function. (i) f(x) 5 – 5 0 x f(x) = ................................................ [2] (ii) f(x) 2 0 x 180° 360° – 2 f(x) = ................................................ [2] (b) f(x) 4 3 P 2 1 – 0.5 0 0.5 1 1.5 2 2.5 x – 1 The diagram shows the graph of another function. By drawing a suitable tangent, find an estimate for the gradient of the function at the point P. ................................................. [3]

Mark scheme: 10(a)(i) x + 5 2 B1 for linear equation with positive gradient or intercept 5 10(a)(ii) 2 sin x oe 2 B1 for recognition of sin or cos(x – 90) 10(b) tangent ruled at P B1 1.3 to 1.4 B2 dep on tangent drawn M1 for rise/run

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2x11 f ( x) = 7 x - 4 g ( x) = , x ! 3 h ( x) = x2 x - 3 (a) Find g(6). ................................................. [1] (b) Find fg(4). ................................................. [2] (c) Find fh(x). ................................................. [1] f ( x) (d) Find + g ( x) . 2 Give your answer as a single fraction, in terms of x, in its simplest form. ................................................. [3] (e) Find the value of x when f ( x + 2) =- 11. x = ................................................ [2] (f) Find the values of p that satisfy h(p) = p. ................................................. [2]

Mark scheme: 11(a) 4 1 11(b) 52 2 2 x M1 for f( 8 ) seen or 7 × − 4 x − 3 11(c) 7x2 – 4 1 11(d) 7 x 2 − 21x + 12 7 x 2 − 21x + 12 3 M1 for ( 7 x − 4 )( x − 3 ) + 2 × 2 x or 2( x − 3) 2 x − 6 B1 for denominator 2 ( x − 3 ) or 2x – 6 final answer 11(e) −3 2 M1 for 7 x + 14 − 4 = −11 11(f) [p =] 0 and [p =] 1 2 B1 for each

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12 (a) A curve has equation y = 4 x 3 - 3x + 3 . (i) Find the coordinates of the two stationary points. ( .................... , .................... ) and ( .................... , .................... ) [5] (ii) Determine whether each of the stationary points is a maximum or a minimum. Give reasons for your answers. [3] (b) The graph of y = x 2 - x + 1 is shown on the grid. y 7 6 5 4 3 2 1 – 3 – 2 – 1 0 1 2 3 4 x – 1 By drawing a suitable line on the grid, solve the equation x 2 - 2x - 2 = 0 . x = .................... or x = .................... [3]

Mark scheme: 12(a)(i)  1   1  5 B2 for 12 x 2 − 3[ = 0]  − , 4  and  , 2   2   2  or B1 for 12x2 or – 3 M1 for their derivative = 0 or dy/dx = 0 B1 for [x =] – ½ and ½ or one coordinate pair correct 12(a)(ii)  1  3 B2 for one correct with reason  − , 4  Max with reason or M1 for correct attempt to find  2  e.g. 2nd derivative/gradients/sketch  1   , 2  Min with reason  2  12(b) line y = x + 3 ruled M2 B1 for [ y = ] x + 3 identified or rules y = x + k or y = px + 3 −0.7 to −0.8 A1 2.7 to 2.8

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