Cambridge IGCSE Mathematics - International 0607 — 2020 May/June Paper 4 · Variant 2

1 A class of 40 students complete a science test. The table shows the marks of the 40 students. Mark 0 1 2 3 4 5 6 7 8 9 10 Number of students 1 1 2 5 5 5 6 3 9 2 1 (a) Write down the mode. .................................................. [1] (b) Work out the range. .................................................. [1] (c) Find the median. .................................................. [1] (d) Find the interquartile range. ................................................. [2] (e) Calculate the mean. ................................................. [2] (f) Two of the students are chosen at random. Find the probability that the difference in their marks is 8. ................................................. [3]

Mark scheme: Question Answer Marks Partial Marks 1(a) 6 1 1(b) 10 1 1(c) 6 1 1(d) 4 2 B1 for LQ = 4 or UQ = 8 1(e) 5.55 2 M1 for attempt at  fx (3 or more terms correct) 1(f) 1 13 3 M2 for two of = or 0.0083 recurring or 120 1560  1 2   2 1   9 1   ×  +  ×  +  ×  0.008333...  40 39   40 39   40 39  or M1 for any one of above products

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2 (a) y 10 9 8 7 6 A 5 4 3 C 2 1 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 x – 1 – 2 – 3 – 4 B – 5 – 6 (i) Describe fully the single transformation that maps triangle A onto triangle B. ............................................................................................................................................. ............................................................................................................................................. [2] (ii) Describe fully the single transformation that maps triangle A onto triangle C. ............................................................................................................................................. ............................................................................................................................................. [3] (b) You may use the grid to help you in answering this question. The transformation P is a rotation of 90° clockwise about the origin. The transformation Q is a reflection in the line y =- x . (i) Find the image of the point ( 5, - 2) under the transformation P. (.................... , ....................) [1] (ii) Find the image of the point ( 5, - 2) under the transformation Q. (.................... , ....................) [1] (iii) Describe fully the single transformation equivalent to P followed by Q. ............................................................................................................................................. ............................................................................................................................................. [2] (iv) Describe fully the single transformation equivalent to Q followed by P. ............................................................................................................................................. ............................................................................................................................................. [2]

Mark scheme: 2(a)(i) Translation 2 B1 for each  − 6     − 10  2(a)(ii) Enlargement 3 B1 for each SF 2 Centre (0, 9) 2(b)(i) (–2, –5) 1 2(b)(ii) (2, –5) 1 2(b)(iii) Reflection 2 B1 for each x-axis oe 2(b)(iv) Reflection 2 B1 for each y-axis oe

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3 Petra is a singer. She wants to estimate how much to spend on advertising. The table shows the amount spent on advertising, $x, and the number of tickets sold, y, for 10 performances. Amount spent ($x) 80 60 50 120 90 40 100 110 70 150 Number of tickets sold ( y) 100 90 60 150 100 75 120 120 100 150 (a) (i) Complete the scatter diagram. The first six points have been plotted for you. y 150 100 Number of tickets sold 50 0 x 0 20 40 60 80 100 120 140 160 Amount spent (dollars) [2] (ii) What type of correlation is shown by the scatter diagram? .................................................. [1] (b) Find the mean amount of money spent on advertising. $ ................................................. [1] (c) (i) Find the equation of the regression line for y in terms of x. y = ................................................. [2] (ii) Use your regression line to estimate the number of tickets sold when Petra spends $130 on advertising. .................................................. [1] (iii) Explain why Petra should not rely on this regression line to estimate the number of tickets she will sell if she spends $500 on advertising. ............................................................................................................................................. ............................................................................................................................................. [1]

Mark scheme: 3(a)(i) All four points correct 2 B1 for two or three correct points 3(a)(ii) positive 1 3(b) 87 1 3(c)(i) [ y = ] 35.9 + 0.811x 2 B1 for 35.9 + kx, or for a + 0.811x, If 0 scored SC1 for 36 + 0.81x 3(c)(ii) 141 1 FT their (i) 3(c)(iii) Outside data range oe 1

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4 A piece of metal is in the shape of a cuboid. The cuboid has length 18 cm, width 12 cm and height 12 cm. A cylinder is removed from the cuboid. The cylinder has length 18 cm and radius 4 cm. NOT TO SCALE 18 cm 12 cm 4 cm 12 cm (a) (i) Find the volume of the metal remaining after the cylinder has been removed. ........................................... cm3 [3] (ii) Write your answer to part (i) in standard form. ........................................... cm3 [1] (b) Find the total surface area of the metal remaining after the cylinder has been removed. ........................................... cm2 [4] (c) The cylinder removed is melted and formed into 16 identical spheres. (i) Calculate the volume of one sphere. ........................................... cm3 [1] (ii) Calculate the radius of one sphere. ............................................. cm [2]

Mark scheme: 4(a)(i) 1690 or 1687 or 1687.1 to 1687.3 3 M2 for 18 × 12 2 − 18 × π × 4 2 or M1 for either term correct 4(a)(ii) 1.69[0] × 10 3 1 FT their (i) 4(b) 1500 or 1503 to 1504 4 M1 for [4 ×]18 × 12 M1 for [2 ×](12 2 − π × 4 2 ) M1 for π × 2 × 4 × 18 4(c)(i) 56.5 or 56.6 or 56.54 to 56.56 1 4(c)(ii) 2.38 or 2.380 to 2.382 2 3 × their ( c ) M1 for 4 × π

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5 Fifty students, 25 boys and 25 girls, were asked which sport they prefer. The results are shown in the table. Athletics Football Swimming Tennis Boy 4 9 2 10 Girl 3 3 12 7 (a) A student is selected at random. Calculate the probability that the student chosen is (i) a girl who prefers swimming, .................................................. [1] (ii) a boy who does not prefer football, .................................................. [1] (iii) a student who prefers athletics. .................................................. [1] (b) Two of the girls are chosen at random. Calculate the probability they both prefer tennis. .................................................. [2] (c) Two of the students who prefer athletics are chosen at random. Calculate the probability that one is a boy and one is a girl. .................................................. [3] (d) Three of the 50 students are chosen at random. Calculate the probability that one is a boy and two are girls and they all prefer swimming. .................................................. [4]

Mark scheme: 5(a)(i) 12 1 oe 50 5(a)(ii) 16 1 oe 50 5(a)(iii) 7 1 oe 50 5(b) 42 2 7 oe M1 for soi 600 25 5(c) 24 3  4 3   3 4  oe M2 for  ×  +  ×  42  7 6   7 6  4 3 3 4 or M1 for × or × 7 6 7 6 5(d) 792 33 4 M3 for or oe 117600 4900  2 12 11   12 2 11   × ×  +  × ×   50 49 48   50 49 48   11 12 2  +  × ×  oe  50 49 48  or M2 for any two products or M1 for any one of above products

More questions on Probability of combined events

6 Herman bought a motorbike on 1 January 2014. By 1 January 2015 the value of the motorbike had reduced by 16%. By 1 January 2016 the value of the motorbike had reduced by 12% of the value on 1 January 2015. The value of the motorbike on 1 January 2016 was $7392. (a) Find how much Herman paid for the motorbike. $ ................................................. [3] (b) From 2016, the value of the motorbike reduced by 8% each year. Calculate the number of complete years it will take for the value of the motorbike to decrease from $7392 to $5000. .................................................. [4]

Mark scheme: 6(a) 10 000 3 7392 M2 for oe (1 − 0.16)(1 − 0.12) or M1 for ÷(1 − 0.16) or ÷ (1 − 0.12) oe or M1 for 88% is ‘equivalent’ to 7392 6(b) 5 4 5000 log 7392 M3 for [ k = ] oe log0.92 or correct trials as far as 4 and 5 k 5000 or M2 for 0.92 = oe 7392 or at least 3 correct trials or M1 for 7392 × 0.92 k = 5000 oe

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7 y 9 x – 6 0 2 – 3 1 (a) f ( )x = 2 + x + 2 (i) On the diagram, sketch the graph of y = f ( x) for values of x between - 6 and 2. [2] (ii) Write down the coordinates of the points where the graph crosses the axes. (............... , ...............) and (............... , ...............) [2] (iii) Write down the equations of the asymptotes of the graph. ........................... , ........................... [2] (b) g ( x) = ( x + 4) 2 On the diagram, sketch the graph of y = g ( x) for - 6 G x G - 1 . [2] (c) Solve the equation. f ( x) = g ( x) ................................................................................................ [3] (d) Solve the inequality. f ( x) H g ( x) ................................................................................................ [2]

Mark scheme: 7(a)(i) 8 y f(x)=2+1/(x+2) 2 B1 for correct ‘hyperbolic shape’ 7 6 B1 for intersects with axes correct 5 4 (approx.) 3 2 1 x -5 -4 -3 -2 -1 1 -1 -2 7(a)(ii) (–2.5, 0) 2 B1 for each (0, 2.5) 7(a)(iii) x = –2 2 B1 for each y = 2 y f(x)=2+1/(x+2)f(x)=(x+4)^2 5 7(b) 4 2 B1 for correct ‘quadratic shape’ 3 B1 for min point at (–4, 0) (approx.) 2 1 -5 -4 -3 -2 -1 1 x -1 -2 -3 -4 -5 7(c) [x =] – 5.30 3 B1 for each correct answer [x =] –3 [x =] –1.70 7(d) −5.30 ≤ x ≤−3 2 B1 for each and −<2 x ≤−1.70

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8 North B 5.37 km NOT TO SCALE North A C 48° 6.13 km 6.42 km D The diagram shows four points A, B, C and D on horizontal ground. B is due North of C and C is due East of A. (a) Find the bearing of (i) D from A, .................................................. [1] (ii) A from D. .................................................. [1] (b) Calculate angle ABC. Angle ABC = ................................................. [2] (c) Calculate the area of quadrilateral ABCD. .......................................... km 2 [3] (d) Calculate CD. CD = ........................................... km [3] (e) Angle ACD is acute. Find the bearing of D from C. .................................................. [4]

Mark scheme: 8(a)(i) 138 1 8(a)(ii) 318 1 FT their (i) + 180 8(b) 48.8 or 48.78… 2 6.13 M1 for tan[ x = ] 5.37 8(c) 31.1 or 31.08… 3 M2 for 6. 13 × 5. 37 1 + × 6. 13 × 6. 42 × sin48 2 2 6.13 × 5. 37 or M1 for or 2 1 × 6. 13 × 6.42 × sin48 2 8(d) 5.11 or 5.111… 3 B2 for 26.1… or M1 for 6.132 + 6.42 2 − 2 × 6.13 × 6.42 × cos48 8(e) 201 or 200.9 to 201.1… 4 B3 for 69[.0] or 68.89 to 68.90 or M2 for sin 48 sin C = × 6.42, [C = 69.0] their (d) sin C sin 48 or M1 for = 6.42 their (d)

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9 f ( )x = 4 - 3 x g ( x) = , x ! 1 h ( )x = x 2 x - 1 (a) Find (i) f ( 2) , .................................................. [1] (ii) f ( g ( 4)) . .................................................. [2] (b) Find g ( g ( - 1)) . .................................................. [2] (c) Solve. h ( f ( x)) = 9 x = .................... or x = .................... [3] (d) Find ( f ( x)) 2 - 1 in terms of x. Give your answer in the form k ( ax + b)( cx + d) where a, b, c, d and k are integers. .................................................. [3]

Mark scheme: 9(a)(i) –2 1 9(a)(ii) 3 2  1  M1 for f   oe  3  9(b) 2 2  1  − oe M1 for g  −  oe 3  2  9(c) 1 7 3 M2 for 4 − 3 x = ± 7 and or M1 for f(x) = ±3 3 3 If 0 scored B1 for each answer 9(d) 3(3 x − 5)( x − 1) 3 B2 for 9 x 2 − 12 x − 12 x + 15 or M1 for (4 − 3 x ) 2 − 1

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10 D NOT TO SCALE 20° 50° A 12 m B C The diagram shows a vertical pole CD. ABC is a straight line on level ground. Find DC. DC = ............................................. m [6]

Mark scheme: 10 6.29 or 6.288 to 6.293 6 12 × tan 20 × tan50 M5 for tan50 − tan 20 12tan20 or M4 for BC = tan50 − tan20 or M3 for BC tan50 = (12 + BC )tan 20 DC or M2 for tan20 = 12 + BC DC and tan50 = BC DC or M1 for tan20 = 12 + BC DC or tan50 = BC

More questions on Trigonometric functions

11 (a) Solve the equations. (i) 5 + 2x = 1 x = ................................................. [2] 10 (ii) 6 - = 1 x x = ................................................. [2] (iii) 3 ( 1 - 2x) = 2 - 4 ( x - 7) x = ................................................. [3] (b) (i) Solve 6x 2 = 7 - 3 x . Give your answers correct to 3 decimal places. You must show all your working. x = .................... or x = .................... [4] (ii) Solve 6y 4 = 7 - 3 y 2 . Give your answers correct to 3 decimal places. y = .................... or y = .................... [2] (c) Solve 2 log x + log 5 = 1. x = ................................................. [4]

Mark scheme: 11(a)(i) –2 2 5 1 M1 for 2 x = 1 − 5 or + x = 2 2 11(a)(ii) 2 2 10 M1 for − = 1 − 6 oe or 6 x − 10 = x x 11(a)(iii) –13.5 3 M1 for correct expansion 3 − 6 x = 2 − 4 x + 28 M1 for correct collection of their terms 3 − 30 = 6 x − 4 x their (3 − 30) M1 for their (6 − 4) 11(b)(i) 6 x 2 + 3 x − 7 = 0 B1 Correct sketch M2 M1 for any U-shaped parabola OR OR b −±3 32 − 4 × 6 × −7 for or b 2 − 4 ac correct 2 a 2 × 6 0.859, –1.359 B1 11(b)(ii) 0.927, 2 FT their (b)(i) –0.927 B1 for each 11(c) 1.41 or 1.414… cao 4 M3 for 5 x 2 = 10 or M2 for log5x 2 [=1] or M1 for logx2 + log5 [=1]

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