Cambridge IGCSE Mathematics 0580 — 2020 Oct/Nov Paper 4 · Variant 2

1 Karel travelled from London to Johannesburg and then from Johannesburg to Windhoek. (a) The flight from London to Johannesburg took 11 hours 10 minutes. The average speed was 813 km/h. Calculate the distance travelled from London to Johannesburg. Give your answer correct to the nearest 10 km. ........................................... km [3] (b) The total time for Karel’s journey from London to Windhoek was 15 hours 42 minutes. The total distance travelled from London to Windhoek was 10 260 km. (i) Calculate the average speed for this journey. ........................................ km/h [2] (ii) The cost of Karel’s journey from London to Windhoek was $470. (a) Calculate the distance travelled per dollar. ........................... km per dollar [1] (b) Calculate the cost per 100 km of this journey. Give your answer correct to the nearest cent. $ .............................. per 100 km [2] (c) Karel changed $300 into 3891 Namibian dollars. Complete the statement. $1 = ............................ Namibian dollars [1]

Mark scheme: Question Answer Marks Partial Marks 1(a) 9080 cao 3 B2 for 9078 to 9081… or M1 for 813 × their 11h 10min 1(b)(i) 654 or 653.5… 2 M1 for 10260 ÷ 15 h 42 min oe 1(b)(ii)(a) 21.8 or 21.82 to 21.83 1 1(b)(ii)(b) 4.58 or 4.59 cao 2 M1 for 470 ÷ (10260 ÷ 100) oe or 100 ÷ their (b)(ii)(a) 1(c) 12.97 1

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2 y 8 7 6 5 4 3 2 T 1 x – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 – 1 – 2 – 3 – 4 P – 5 – 6 – 7 – 8 (a) Describe fully the single transformation that maps triangle T onto triangle P. ..................................................................................................................................................... ..................................................................................................................................................... [2] (b) (i) Reflect triangle T in the line x = 1. [2] (ii) Rotate triangle T through 90° anticlockwise about (6, 0). [2] (iii) Enlarge triangle T by a scale factor of -2, centre (1, 0). [2]

Mark scheme: 2(a) Translation 2 B1 for each  1     − 6  2(b)(i) Image at (0, 1), (–3, 1), (–3, 2) 2 B1 for reflection in x = k or y = 1 2(b)(ii) Image at (5, –4), (5, –1), (4, –1) 2 B1 for rotation 90° anticlockwise with other centre or for rotation 90° clockwise about (6, 0) 2(b)(iii) Image at (–1, –2), (–7, –2), (–7, –4) 2 B1 for enlargement, factor –2 with other centre

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3 (a) Beth invests $2000 at a rate of 2% per year compound interest. (i) Calculate the value of this investment at the end of 5 years. $ ................................................ [2] (ii) Calculate the overall percentage increase in the value of Beth’s investment at the end of 5 years. ............................................. % [2] (iii) Calculate the minimum number of complete years it takes for the value of Beth’s investment to increase from $2000 to more than $2500. ................................................. [3] (b) The population of a village decreases exponentially at a rate of 4% each year. The population is now 255. Calculate the population 16 years ago. ................................................. [3]

Mark scheme: 3(a)(i) 2210 or 2208 or 2208.2, or 2208.16… 2 5  2  M1 for 2000 ×  1 +  oe  100  3(a)(ii) 10.4 or 10.5 or 10.40 to 10.41 2 their(a)(i) − 2000 M1 for [×100] or 2000 their(a)(i)  2  5 ×100 or  1 +  – 1 or 2000  100   2  5  1 +  × 100 oe  100  3(a)(iii) 12 3 B2 for 11.3 or 11.26 to 11.27 OR  2 11 M2 for [2000 ×]  1 +  oe  100   2 12 or [2000 ×]  1 +  oe seen  100   2  n or M1 for [2000 ×]  1 +  oe, n > 5 oe  100   2  n or for 2000 ×  1 +  = or > or ⩾ 2500  100  oe 3(b) 490 cao 3 16  4  M2 for p ×  1 −  = 255 oe soi by  100  490.0...  4  n or M1 for p ×  1 −  = 255 oe,  100  n > 1 oe

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4 The height, h cm, of each of 120 plants is measured. The cumulative frequency diagram shows this information. 120 100 80 Cumulative 60 frequency 40 20 0 0 10 20 30 40 50 h Height (cm) (a) Use the cumulative frequency diagram to find an estimate of (i) the median, ............................................ cm [1] (ii) the interquartile range, ............................................ cm [2] (iii) the 60th percentile, ............................................ cm [1] (iv) the number of plants with a height greater than 40 cm. ................................................. [2] (b) The information in the cumulative frequency diagram is shown in this frequency table. Height, h cm 0 1 h G 10 10 1 h G 20 20 1 h G 30 30 1 h G 50 Frequency 2 18 62 38 (i) Calculate an estimate of the mean height. ............................................ cm [4] (ii) A histogram is drawn to show the information in the frequency table. The height of the bar representing the interval 10 1 h G 20 is 7.2 cm. Calculate the height of the bar representing the interval 30 1 h G 50 . ............................................ cm [2]

Mark scheme: 4(a)(i) 25 1 4(a)(ii) 10 nfww 2 B1 for [lq =] 22 or [uq =] 32 4(a)(iii) 27 1 4(a)(iv) 6 2 B1 for 114 written 4(b)(i) 27.9 or 27.91 to 27.92 nfww 4 M1 for mid-values M1 where x lies within or on forfx boundary of correct interval M1 dep ÷ 120 dep on second M1 fx 4(b)(ii) 7.6 2 18 38 M1 for oe or oe 10 20 or B1 for [multiplier] 4 or ¼

More questions on Histograms

5 Ahmed sells different types of cake in his shop. The cost of each cake depends on its type and its size. Every small cake costs $x and every large cake costs $(2x + 1). (a) The total cost of 3 small lemon cakes and 2 large lemon cakes is $12.36 . Find the cost of a small lemon cake. $ ................................................ [3] (b) The cost of 18 small chocolate cakes is the same as the cost of 7 large chocolate cakes. Find the cost of a small chocolate cake. $ ................................................ [3] (c) The number of small cherry cakes that can be bought for $4 is the same as the number of large cherry cakes that can be bought for $13. Find the cost of a small cherry cake. $ ................................................ [3] (d) Petra spends $20 on small coffee cakes and $10 on large coffee cakes. The total number of cakes is 45. Write an equation in terms of x. Solve this equation to find the cost of a small coffee cake. Show all your working. $ ................................................ [7]

Mark scheme: 5(a) 1.48 3 B2 for 7x + 2 = 12.36 or better or M1 for 3x + 2(2x + 1) [= 12.36] or better 5(b) 3 3 B2 for 18x – 14x = 7 or better 1.75 or 1 or M1 for 18x = 7(2x + 1) 4 5(c) [0].8 oe 3 B2 for 4(2x + 1) = 13x 4 13 or M1 for = oe x 2 x + 1 or correct equation to find number of cakes 5(d) 20 10 M2 20 10 + = 45 oe B1 for seen or seen x 2 x + 1 x 2 x + 1 90x2 – 5x – 20 [= 0] oe B2 20(2 x + 1) + 10 x B1 for = 45 or better x (2 x + 1) (9x + 4)(2x – 1) [= 0] or for M2 FT their 3-term quadratic 2 M1 for factors that give two correct terms −−±1 ( −1) − 4(18)( −4) oe when expanded 2(18) −b or for correct discriminant or correct 2 a provided quadratic formula is in correct form 1 B1 [0].5 or final answer 2

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6 4 6 3 4 2 3 Red Yellow Blue Blue Yellow Blue The diagram shows six discs. Each disc has a colour and a number. (a) One disc is picked at random. Write down the probability that (i) the disc has the number 4, ................................................. [1] (ii) the disc is red and has the number 3, ................................................. [1] (iii) the disc is blue and has the number 4. ................................................. [1] (b) Two of the six discs are picked at random without replacement. Find the probability that (i) both discs have the number 3, ................................................. [2] (ii) both discs have the same colour. ................................................. [3] (c) Two of the six discs are picked at random with replacement. Find the probability that both discs have the same colour. ................................................. [3]

Mark scheme: 6(a)(i) 1 1 oe 3 6(a)(ii) 0 1 6(a)(iii) 1 1 oe 6 6(b)(i) 1 2 2 1 oe M1 for × or equivalent method 15 6 5 6(b)(ii) 4 3 2 1 3 2 oe M2 for × + × or equivalent method 15 6 5 6 5 2 1 3 2 or M1 for × oe seen or × oe seen 6 5 6 5 6(c) 7 3  1  2  2  2  3  2 oe M2 for + + oe       18  6   6   6  or M1 for one correct product seen or sample space with 14 correct pairs identified

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2 17 y = x + , x ! 0 x (a) Complete the table. x 0.2 0.3 0.5 1 1.5 2 2.5 y 5.0 3.4 2.3 2.9 6.7 [2] 2 1 (b) On the grid, draw the graph of y = x + for 0.2 G x G 2.5 . x 2 1 The graph of y = x + for - 2.5 G x G - 0.2 has been drawn for you. x y 7 6 5 4 3 2 1 – 2.5 – 2 – 1.5 – 1 – 0.5 0 0.5 1 1.5 2 2.5 x – 1 – 2 – 3 – 4 – 5 [4] (c) By drawing suitable straight lines on the grid, solve the following equations. 2 1 (i) x + =- 2 x x = ................................................ [1] 2 1 (ii) x + + x - 1 = 0 x x = ................................................ [2] 2 1(d) k is an integer and the equation x + = k has three solutions. x Write down a possible value of k. k = ................................................ [1]

Mark scheme: 7(a) 2, 4.5 2 B1 for each 7(b) Correct graph 4 B3 FT for 6 or 7 correct points FT their table or B2 FT for 4 or 5 correct points FT their 6666 table 4444 or B1 FT for 2 or 3 correct points FT their 2222 table -2-2-2-2 -1-1-1-1 0000 0000 1111 2222 -2-2-2-2 -4-4-4-4 7(c)(i) –0.5 to –0.4 1 7(c)(ii) y = 1 – x ruled 2  2 1  M1 for [y =] 1 – x or x + = 1 − x soi and    x  –1.9 to –1.75 or B1 for –1.9 to –1.75 7(d) Any integer > 2 1

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8 (a) D x° z° E NOT TO SCALE P 80° C v° y° w° 40° A B Q The points A, B, C, D and E lie on the circle. PAQ is a tangent to the circle at A and EC = EB. Angle ECB = 80° and angle ABE = 40°. Find the values of v, w, x, y and z. v = ................... w = ................... x = ................... y = ................... z = ................... [5] (b) NOT TO SCALE O M K L In the diagram, K, L and M lie on the circle, centre O. Angle KML = 2x° and reflex angle KOL = 11x°. Find the value of x. x = ................................................ [3] (c) D NOT TO SCALE C X A B The diagonals of the cyclic quadrilateral ABCD intersect at X. (i) Explain why triangle ADX is similar to triangle BCX. Give a reason for each statement you make. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. [3] (ii) AD = 10 cm, BC = 8 cm, BX = 5 cm and CX = 7 cm. (a) Calculate DX. DX = ........................................... cm [2] (b) Calculate angle BXC. Angle BXC = ................................................ [4]

Mark scheme: 8(a) [v = ] 40 5 B1 for each [w = ] 80 FT angle z as 140 – their w [x = ] 40 [y = ] 100 [z = ] 60 8(b) 24 3 M2 for 360 – 11x = 2 × 2x oe or M1 for 360 – 11x seen or obtuse angle KOL = 2 × 2x oe 8(c)(i) angle ADX = angle BCX oe M2 Accept in any order same segment oe M1 for one correct pair with reason angle DAX = angle CBX oe If 0 scored, SC1 for two correct pairs of same segment oe equal angles identified with incorrect/no reasons angle AXD = BXC oe [vertically] opposite oe corresponding angles are equal oe A1 8(c)(ii)(a) 8.75 or 8¾ 2 8 7 M1 for = oe 10 DX 8(c)(ii)(b) 81.8 or 81.78 to 81.79 4 5 2 + 7 2 − 8 2 M2 for [cos[BXC] =] oe 2 × 5 × 7 or M1 for 8 2 = 5 2 + 7 2 −×2 5 × 7 × cos(...) oe 10 A1 for oe 70

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9 H 5 cm G NOT TO D SCALE C E F 70° 12 cm A B 8 cm The diagram shows a prism with a rectangular base, ABFE. The cross-section, ABCD, is a trapezium with AD = BC. AB = 8 cm, GH = 5 cm, BF = 12 cm and angle ABC = 70°. (a) Calculate the total surface area of the prism. .......................................... cm2 [6] (b) The perpendicular from G onto EF meets EF at X. (i) Show that EX = 6.5 cm. [1] (ii) Calculate AX. AX = ........................................... cm [2] (iii) Calculate the angle between the diagonal AG and the base ABFE. ................................................. [2]

Mark scheme: 9(a) 315 or 314.5 to 315.0 6 height M1 for tan70 = oe or better seen 1 ( 8 - 5 ) 2 1 M1dep for ( 8 + 5 ) ×their height or better 2 seen dep on trig attempt for height 1 ( 8 − 5 ) 2 M2 for 12 × oe or better seen cos70 1 ( 8 − 5 ) 2 or M1 for oe or better seen cos70 M1 for 8 × 12 oe isw and 5 × 12 oe isw 9(b)(i) 8 – ½ (8 – 5) or 5 + ½ (8 – 5) M1 9(b)(ii) 13.6 or 13.64 to 13.65 2 M1 for 122 + (6.5)2 oe 9(b)(iii) 16.8 or 16.9 or 16.79 to 16.91… 2 M1 for identifying angle GAX from a nfww diagram or from working or better

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2 110 f ( x) = x + 1 g ( x) = 1 - 2 x h ( x) = , x ! 0 j ( x) = 5x x (a) Find the value of (i) f(3), ................................................. [1] (ii) gf(3). ................................................. [1] (b) Find g -1 ( x) . g -1 ( x) = ................................................ [2] (c) Find x when h ( x) = 2 . x = ................................................ [1] (d) Find g(x)g(x) - gg(x), giving your answer in the form ax 2 + bx + c . ................................................. [4] (e) Find hh(x), giving your answer in its simplest form. ................................................. [1] (f) Find j(5). ................................................. [1] (g) Find x when j -1 ( x) = 2 . x = ................................................ [1] (h) j ( x) = hg ( - 12) Find the value of x. x = ................................................ [2] Question 11 is printed on the next page.

Mark scheme: 10(a)(i) 10 1 10(a)(ii) –19 1 FT 1 – 2 their (a)(i) 10(b) 1 −x 2 y 1 oe final answer M1 for x = 1 – 2y or y + 2x = 1 or = − x 2 2 2 or y – 1 = –2x or better 10(c) 1 1 oe 2 10(d) 4x2 – 8x + 2 final answer 4 M1 for (1 – 2x)(1 – 2x) – (1– 2(1 – 2x)) or better B1 for 1 – 2x – 2x + 4x2 B1 for – (1 – 2 + 4x) or better or [+] 1 – 4x or for correct answer seen then spoiled 10(e) x final answer 1 10(f) 3125 1 10(g) 25 1 10(h) –2 2 1 B1 for or 0.04 25

More questions on Functions

11 Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 13 9 5 1 B 0 7 26 63 7 8 9 10 C 8 16 32 64 (a) Complete the table for the three sequences. [10] (b) One term in Sequence C is p. q Write down the next term in Sequence C in terms of p and q. ................................................. [2]

Mark scheme: 11(a) A : –3 17 – 4n oe 3 B1 for –3 B2 for 17 – 4n oe or B1 for k – 4n oe or 17 – pn oe, p ≠ 0 B : 124 n3 – 1 oe 3 B1 for 124 B2 for n3 – 1 oe or B1 for any cubic 11 n + 6 4 11 C : oe B1 for 128 2 n + 2 128 n + 6 B3 for oe 2 n + 2 or B2 for 2 n + 2 oe seen or B1 for 2k oe or n + 6 seen 11(b) p + 1 2 B1 for p + 1 or 2q oe oe 2 q

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