Cambridge IGCSE Mathematics 0580 — 2020 May/June Paper 4 · Variant 2

1 (a) (i) Divide $24 in the ratio 7 : 5. $ ................... , $ ................... [2] (ii) Write $24.60 as a fraction of $2870. Give your answer in its lowest terms. .................................................. [2] (iii) Write $1.92 as a percentage of $1.60 . ............................................. % [1] (b) In a sale the original prices are reduced by 15%. (i) Calculate the sale price of a book that has an original price of $12. $ ................................................. [2] (ii) Calculate the original price of a jacket that has a sale price of $38.25 . $ ................................................. [2] (c) (i) Dean invests $500 for 10 years at a rate of 1.7% per year simple interest. Calculate the total interest earned during the 10 years. $ ................................................ [2] (ii) Ollie invests $200 at a rate of 0.0035% per day compound interest. Calculate the value of Ollie’s investment at the end of 1 year. [1 year = 365 days.] $ ................................................ [2] (iii) Edna invests $500 at a rate of r % per year compound interest. At the end of 6 years, the value of Edna’s investment is $559.78 . Find the value of r. r = ................................................ [3]

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 14, 10 2 M1 for 24 ÷ (7 + 5) 1(a)(ii) 3 2 B1 for correct fraction not in lowest terms 350 1(a)(iii) 120 1 1(b)(i) 10.2[0] 2 15 M1 for × 12 oe or better 100 1(b)(ii) 45 2 38.25 M1 for oe 15 1 − 100 1(c)(i) 85 2 500 × 1.7 × 10 M1 for oe 100 1(c)(ii) 203 or 202.5 to 202.6 2 365  0.0035  M1 for 200 ×  1 +   100  1(c)(iii) 1.9 3 559.78 M2 for 6 500  r 6 or M1 for 500  1 +  = 559.78  100 

More questions on Exponential growth and decay

4 - 2 2 (a) p = q = e 5 o e 7o (i) Find 2 p + q . [2] f p (ii) Find p . ................................................. [2] - 3 (b) A is the point (4, 1) and AB = e 1o. Find the coordinates of B. ( ...................... , ...................... ) [1] (c) The line y = 3 x - 2 crosses the y-axis at G. Write down the coordinates of G. ( ...................... , ...................... ) [1] (d) D NOT TO T SCALE M O C In the diagram, O is the origin, OT = 2TD and M is the midpoint of TC. OC = c and OD = d . Find the position vector of M. Give your answer in terms of c and d in its simplest form. ................................................. [3]

Mark scheme: 2(a)(i)  6  2 B1 for each    17  2(a)(ii) 6.4[0] or 6.403... 2 M1 for 42 + 52 2(b) (1, 2) 1 2(c) (0, –2) 1 2(d) 1 1 3 B2 for correct unsimplified answer c + d  2 2 3 or M1 for CT = – c + d oe 3  2 or TC = c – d oe 3 or for correct route

More questions on Vectors in two dimensions

3 The speed, v km/h, of each of 200 cars passing a building is measured. The table shows the results. Speed (v km/h) 0 1 v G 20 20 1 v G 40 40 1 v G 45 45 1 v G 50 50 1 v G 60 60 1 v G 80 Frequency 16 34 62 58 26 4 (a) Calculate an estimate of the mean. ........................................ km/h [4] (b) (i) Use the frequency table to complete the cumulative frequency table. Speed (v km/h) v G 20 v G 40 v G 45 v G 50 v G 60 v G 80 Cumulative frequency 16 50 196 200 [1] (ii) On the grid, draw a cumulative frequency diagram. 200 180 160 140 120 Cumulative frequency 100 80 60 40 20 0 v 0 10 20 30 40 50 60 70 80 Speed (km/h) [3] (iii) Use your diagram to find an estimate of (a) the upper quartile, ........................................ km/h [1] (b) the number of cars with a speed greater than 35 km/h. ................................................. [2] (c) Two of the 200 cars are chosen at random. Find the probability that they both have a speed greater than 50 km/h. ................................................. [2] (d) A new frequency table is made by combining intervals. Speed (v km/h) 0 1 v G 40 40 1 v G 50 50 1 v G 80 Frequency 50 120 30 On the grid, draw a histogram to show the information in this table. 15 10 Frequency density 5 0 v 0 10 20 30 40 50 60 70 80 Speed (km/h) [3]

Mark scheme: 3(a) 41.4 4 M1 for 10, 30, 42.5, 47.5, 55, 70 M1 for Σ fx where x lies in or on the boundary of each interval. Σfx M1 dep for dep on second M1 200 3(b)(i) 112, 170 1 3(b)(ii) Correct diagram 3 B1 for correct horizontal plot B1FT for correct vertical plots B1 FT dep on at least B1 earned for reasonable increasing curve or polygon through their 6 points If 0 scored SC1FT for 5 out of 6 points plotted correctly 3(b)(iii)(a) 48 1 3(b)(iii)(b) 160 2 M1 for 40 seen 3(c) 87 2 30 29 oe M1 for × oe 3980 200 199 3(d) Correct histogram 3 B1 for each column If 0 scored SC1 for correct frequency densities soi 1.25, 12, 1

More questions on Probability of combined events

4 S NOT TO SCALE 55° P 150 m 25° 45° R 120 m Q The diagram shows two triangles. (a) Calculate QR. QR = ............................................ m [3] (b) Calculate RS. RS = ............................................ m [4] (c) Calculate the total area of the two triangles. ............................................ m2 [3]

Mark scheme: 4(a) 65.4 or 65.36 to 65.37 3 M1 for 1502 + 1202 – 2 × 150 × 120 cos 25 A1 for 4270 or 4272 to 4273 4(b) 125 or 124.7 to 124.8 4 B1 for [angle S =] 80 150sin55 M2 for sin their 80 sin their 80 sin55 or M1 for = oe 150 RS 4(c) 10 400 or 10 410 to 10 440 nfww 3 1 M1 for × 120 × 150sin25 oe 2 1 M1 for × 150 × their (b) sin45 oe 2

More questions on Non-right-angled triangles

5 North D NOT TO SCALE 140° A 450 m 400 m B 350 m C The diagram shows a field ABCD. The bearing of B from A is 140°. C is due east of B and D is due north of C. AB = 400 m, BC = 350 m and CD = 450 m. (a) Find the bearing of D from B. ................................................. [2] (b) Calculate the distance from D to A. ............................................. m [6] (c) Jono runs around the field from A to B, B to C, C to D and D to A. He runs at a speed of 3 m/s. Calculate the total time Jono takes to run around the field. Give your answer in minutes and seconds, correct to the nearest second. .................. min .................. s [4]

Mark scheme: 5(a) [0]38 or [0]37.9 or [0]37.87... 2 350 M1 for tan = oe 450 If 0 scored, SC1 for answer [0]52 or [0]52.1 or [0]52.12 to [0]52.13 5(b) 624 or 623.8 to 623.9 6 M2 for 450 – 400 sin 50 ... or M1 for sin 50 = 400 M2 for 350 + 400 cos 50 ... or M1 for cos 50 = 400 M1 for (their (450 – 400 sin 50))2 + (their (350 + 400 cos 50))2 5(c) 10 min 8 s 4 B3 for 10.1 or 10.13… or M2 for (400 + 350 + 450 + their DA) ÷ 3 [÷ 60] oe or M1 for any distance ÷ 3 M1 for rounding their minutes into minutes and seconds to nearest second if clearly seen

More questions on Geometrical terms

6 f ( x) = 3x + 2 g ( x) = x 2 + 1 h ( x) = 4x (a) Find h(4). ................................................. [1] (b) Find fg(1). ................................................. [2] (c) Find gf(x) in the form ax 2 + bx + c . ................................................. [3] (d) Find x when f ( x) = g ( 7) . x = ................................................ [2] (e) Find f -1 ( x) . f -1 ( x) = ................................................ [2] g (x) (f) Find + x . f (x) Give your answer as a single fraction, in terms of x, in its simplest form. ................................................. [3] (g) Find x when h -1 ( x) = 2 . x = ................................................ [1]

Mark scheme: 6(a) 256 1 6(b) 8 2 M1 for 3(x2 + 1) + 2 or for 3(2) + 2 6(c) 9 x 2 + 12 x + 5 3 M1 for (3x + 2)2 + 1 B1 for [(3x + 2)2 =] 9 x 2 + 6 x + 6 x + 4 oe 6(d) 16 2 M1 for 3x + 2 = 72 + 1 or better 6(e) x− 2 2 M1 for x = 3y + 2 or for y – 2 = 3x or for oe final answer y 2 3 = x + 3 3 6(f) 4 x 2 + 2 x + 1 3 B1 for x2 + 1 + x (3x + 2) or better seen final answer M1 for common denominator 3x + 2 3 x + 2 6(g) 16 1

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7 Tanya plants some seeds. The probability that a seed will produce flowers is 0.8 . When a seed produces flowers, the probability that the flowers are red is 0.6 and the probability that the flowers are yellow is 0.3 . (a) Tanya has a seed that produces flowers. Find the probability that the flowers are not red and not yellow. ................................................. [1] (b) (i) Complete the tree diagram. Produces Colour flowers Red ............... ............... Yes Yellow 0.8 ............... Other colours No ............... [2] (ii) Find the probability that a seed chosen at random produces red flowers. ................................................. [2] (iii) Tanya chooses a seed at random. Find the probability that this seed does not produce red flowers and does not produce yellow flowers. ................................................. [3] (c) Two of the seeds are chosen at random. Find the probability that one produces flowers and one does not produce flowers. ................................................. [3]

Mark scheme: 7(a) 0.1 1 7(b)(i) 0.2 oe 2 B1 for 0.2 0.6, 0.3, 0.1 oe B1 for 0.6, 0.3, 0.1 7(b)(ii) 0.48 oe 2 FT their 0.6 from tree diagram M1 for 0.8 × their 0.6 7(b)(iii) 0.28 oe 3 M2 for 0.2 + 0.8 × 0.1 oe or M1 for 0.2 or 0.8 × 0.1 or 0.8 × (0.6 + 0.3) 7(c) 0.32 oe 3 M2 for 0.8 × 0.2 + 0.2 × 0.8 oe M1 for one of these products

More questions on Probability of combined events

8 (a) C R NOT TO SCALE A B 8 cm P Q 12 cm Triangle ABC is mathematically similar to triangle PQR. The area of triangle ABC is 16 cm2. (i) Calculate the area of triangle PQR. .......................................... cm2 [2] (ii) The triangles are the cross-sections of prisms which are also mathematically similar. The volume of the smaller prism is 320 cm3. Calculate the length of the larger prism. ............................................ cm [3] (b) A cylinder with radius 6 cm and height h cm has the same volume as a sphere with radius 4.5 cm. Find the value of h. 4 3 [The volume, V, of a sphere with radius r is V = rr . ] 3 h = ................................................ [3] (c) A solid metal cube of side 20 cm is melted down and made into 40 solid spheres, each of radius r cm. Find the value of r. 4 3 [The volume, V, of a sphere with radius r is V = rr . ] 3 r = ................................................ [3] 7x(d) A solid cylinder has radius x cm and height cm. 2 The surface area of a sphere with radius R cm is equal to the total surface area of the cylinder. Find an expression for R in terms of x. [The surface area, A, of a sphere with radius r is A = 4rr 2 . ] R = ................................................ [3]

Mark scheme: 8(a)(i) 36 2 2 2  8   12  M1 for   or   oe  12   8  8(a)(ii) 30 3 12 M2 for 320 ÷ 16 × oe 8 or M1 for 320 ÷ 16 8(b) 3.375 cao 3 4 3 π × 4.5 3 M2 for or better π × 6 2 2 4 3 or M1 for π × 6 × h = × π × 4.5 3 8(c) 3.63 or 3.627 to 3.628 3 20 3 M2 for 4 40 × π 3 4 3 3 or M1 for 40 × × π × r = 20 3 8(d) 3x 1 3 B2 for 4 R 2 = 9 x 2 oe or better or 1.5x or x 2 12 2 2 7 x or M1 for 4πR = 2πx + π × 2 x × 2

More questions on Surface area and volume

9 (a) (i) Write x 2 + 8x - 9 in the form ( x + k) 2 + h . ................................................. [2] (ii) Use your answer to part (a)(i) to solve the equation x 2 + 8x - 9 = 0 . x = ................... or x = ................... [2] 2 - 7 + 61 - 7 - 61 (b) The solutions of the equation x + bx + c = 0 are and . 2 2 Find the value of b and the value of c. b = ................................................ c = ................................................ [3] (c) (i) y O x On the diagram, (a) sketch the graph of y = ( x - 1) 2 , [2] 1 (b) sketch the graph of y = x + 1. [2] 2 2 1 (ii) The graphs of y = ( x - 1) and y = x + 1 intersect at A and B. 2 Find the length of AB. AB = ................................................ [7] Question 10 is printed on the next page.

Mark scheme: 9(a)(i) 2 2 2 2 2 ( x + 4) − 25 B1 for ( x + k ) −−9 (theirk ) or ( x + 4) − h or k = 4 9(a)(ii) x + 4 = [ ± ] 5 M1 FT their (a)(i) –9 and 1 A1 9(b) [b =] 7 3 B1 for [b = ] 7 [c =] –3 M1 for b2 – 4c = 61 9(c)(i)(a) Correct sketch 2 B2 for correct quadratic curve with min touching x-axis 8888 or B1 for parabola vertex downwards 6666 4444 2222 -2-2-2-2 00000000 2222 4444 9(c)(i)(b) Correct sketch 2 B2 for correct straight line intersecting curve on 6666 y-axis 5555 or B1 for straight line with positive gradient and 4444 positive y-intercept 3333 2222 1111 4444 -3-3-3-3 -2-2-2-2 -1-1-1-1 00000000 1111 2222 -1-1-1-1 9(c)(ii) 2.8[0] or 2.795... 7 2 5 B3 for x − x = 0 oe 2 2 1 or M1 for ( x − 1) = x + 1 2 B1 for [(x – 1)2 =] x2 – x – x + 1 AND  5 9  B2 for (0, 1) and  ,  oe  2 4  5 or B1 [x =] 0 and oe 2 AND M1 for (difference in x )2 + (difference in y)2

More questions on Equations

10 (a) y = x 4 - 4x 3 (i) Find the value of y when x =- 1. y = ................................................ [2] (ii) Find the two stationary points on the graph of y = x 4 - 4x 3 . ( ..................... , ..................... ) ( ..................... , ..................... ) [6] (b) y = x p + 2x q d y 10 4 d y = 11x + 10x , where is the derived function. d x d x Find the value of p and the value of q. p = ................................................ q = ................................................ [2]

Mark scheme: 10(a)(i) 5 2 M1 for (–1)4 – 4(–1)3 10(a)(ii) (0, 0) and (3, –27) 6 B2 for 4x3 – 12x2 [ = 0] or B1 for 4x3 or 12x2 AND M1 for derivative = 0 or their derivative = 0 M1 for 4x2(x – 3)[= 0] B1 for [x =] 0 and [ x =] 3 or [y =] 0 and [y =] –27 or for one correct coordinate pair 10(b) [p =] 11 2 B1 for each [q =] 5 dy p −1 q −1 or M1 for = px + 2qx dx

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