Cambridge IGCSE Mathematics - International 0607 — 2023 May/June Paper 4 · Variant 3

1 y 8 7 6 5 4 3 A 2 B 1 – 8 – 7 – 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4 5 6 7 8 x – 1 – 2 – 3 – 4 – 5 – 6 – 7 – 8 - 6 (a) Translate triangle A by Label the image C. [2] e 4o. (b) Describe fully the single transformation that maps triangle A onto triangle B. ..................................................................................................................................................... ..................................................................................................................................................... [3] (c) Reflect triangle A in the line x =- 2 . Label the image D. [2] (d) Enlarge triangle A by scale factor - 1, centre (0, 0). Label the image E. [2]

Mark scheme: Question Answer Marks Partial Marks 1(a) Correct triangle 2  6  k B1 for translation by   or  (–5, 5) (–5, 7) (–6, 7)  k  4 1(b) Rotation 3 B1 for each 90 clockwise oe Centre (2, 0) 1(c) Correct triangle 2 B1 for correct reflection in x = k or in y (–5, 1) (–5, 3) (–4, 3) = –2 1(d) Correct triangle 2 B1 for correct enlargement, incorrect (–1, –1) (0, –3) (–1, –3) centre

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2 A triathlon race consists of three parts: • a 1500 m swim • a 40 km bike ride • a 10 km run. (a) John swims the 1500 m in 25 minutes. Find his average speed, in km/h, for this swim. ......................................... km/h [2] (b) John completes the 40 km bike ride at an average speed of 32 km/h. Find the time, in minutes, for John to complete this bike ride. ........................................... min [2] (c) John completes the whole race at an average speed of 20.6 km/h. Find the average speed, in km/h, for John to complete his 10 km run. ......................................... km/h [3]

Mark scheme: 2(a) 3.6 oe 2 1500 1.5 25 M1 for or or figs15  25 25 60 2(b) 75 2 40 M1 for 32 2(c) 12 3 M1 for 51.5  20.6 oe M1 for 10   their150  25  their (b )  [ 60] oe

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3 The table shows the marks of 12 students in a French examination and a Spanish examination. French mark (x) 17 23 28 32 37 42 57 61 77 82 94 96 Spanish mark (y) 26 22 33 46 41 53 62 67 66 75 83 95 (a) Find the median Spanish mark. ................................................. [1] (b) Find the mean French mark. ................................................. [1] (c) Find the equation of the regression line for y in terms of x. y = ................................................. [2] (d) Use your equation to estimate the Spanish mark when (i) the French mark is 50 ................................................. [1] (ii) the French mark is 6. ................................................. [1] (e) Which French mark, 50 or 6, is likely to give the most reliable Spanish mark? Give a reason for your answer. .................... because .................................................................................................................. ..................................................................................................................................................... [1]

Mark scheme: 3(a) 57.5 1 3(b) 53.8 or 53.83… 1 3(c) 0.791x + 13.2 2 M1 for 0.791x + k or kx + 13.2 or 0.79x + 13 3(d)(i) 53 or 52.7 to 52.8 1 FT their part (b) 3(d)(ii) 18 or 17.90 to 17.95 1 FT their part (b) 3(e) 50, within range 1 or 50 as 6 is not in range

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4 (a) $x is divided in the ratio 3 : 5. The larger share is $42. Find the value of x. x = ................................................. [2] (b) (i) Increase 124 by 16%. ................................................. [2] 2 (ii) The price of a coat is reduced by in a sale. 9 The new price of the coat is $73.50 . Find the original price of the coat. $ ................................................. [2] (c) Xiong invests $2000 in Bank A which pays simple interest at a rate of 3% each year. Find the total amount of interest Xiong receives at the end of 5 years. $ ................................................. [2] (d) Wendi invests $400 in Bank B which pays compound interest at a rate of 1.6% each year. Find the total amount of interest Wendi receives at the end of 3 years. $ ................................................. [3] (e) Pedro invests $1000 in Bank C for 18 years. Pedro also invests $1000 in Bank D for 18 years. Bank C pays simple interest at a rate of x % each year. Bank D pays compound interest at a rate of 0.7x % each year. At the end of 18 years Pedro has exactly the same amount of money in Bank C and Bank D. 18 18x 0 .7 x (i) Show that 1 + = 1 + . 100 e 100 o [2] (ii) Given that 5 1 x 1 7 , use a graphical method to find x. x 5 7 x = ................................................. [3]

Mark scheme: 4(a) 67.2[0] 2 M1 for 42  5  k  where k is 8 or 3 or 1 4(b)(i) 143.84 cao 2 16 M1 for 124  (1  ) oe 100 or B1 for 19.84 or 143.84 seen 4(b)(ii) 94.5 [0] 2  2  M1 for  1   x  73.5 oe  9  4(c) 300 2 M1 for 2000  [0].03  5 oe 4(d) 19.51 3 3  1.6  M2 for 400  1    400 oe  100   1.6 3 or M1 for 400  1   oe  100  4(e)(i) 1000  x  18 M1 1000  oe 100 18 M1  0.7 x  1000  1   oe  100  4(e)(ii) 5.76 or 5.756... 3  18 x  M2 for sketch of [ y ]  1   and  100   0.7 x 18 [ y  ]  1   oe  100  with distinct curve and line with clear point of intersection  18 x  or M1 for sketch of [ y ]  1   or  100   0.7 x 18 [ y  ]  1   oe  100 

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5 B NOT TO 50 cm SCALE 55 cm A 64 cm C D E In the diagram, AB is parallel to ED. ACD and BCE are straight lines. AB = 50 cm, BC = 55 cm and AC = 64 cm . (a) Show that angle ACB = 49.0° correct to one decimal place. [3] (b) Use the sine rule to calculate angle CAB. Angle CAB = ................................................. [3] (c) Calculate the area of triangle ABC. .......................................... cm2 [2] 2(d) AC = AD 3 Calculate the area of triangle CDE. .......................................... cm2 [2]

Mark scheme: 5(a) 55 2  64 2  50 2 M2 M1 for cos ACB  2 2 2 2  55  64 50  55  64 2 55  64  cos ACB  ACB   48.97...   49.0  A1 5(b) 55sin49 55 50 sin CAB  M2 M1 for  oe 50 sin CAB sin 49 56.1 or 56.07 to 56.12 B1 5(c) 1330 or 1327 to 1328… 2 M1 for 0.5  64  55  sin49 oe 5(d) 331.9 to 333 2 FT their 5(c) ÷ 4 M1 for 0.5 2 or 2 2 1 or  32  27.5sin 49.0 oe 2

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6 (a) Solve. 7x - 5 = 3x + 13 x = ................................................. [2] (b) Solve. 4 ( 2x - 3) = 3 ( 1 - 2 x) x = ................................................. [3] (c) Solve. 3x + 2 2 = 8 3x + 2 x = .................. or x = .................. [3] (d) Solve. 1 - 2 x 2 = 5x - 1 Give your answer correct to two decimal places. x = .................. or x = .................. [3] (e) log x = 1 + 4 log y Find x in terms of y. x = ................................................. [3] (f) There are 12 balls in a bag, n of them are blue. A ball is taken from the bag at random and replaced. The probability that the ball is blue is p. 6 more blue balls are added to the bag. A ball is taken from the bag at random. The probability that this ball is blue is 2p. Find the value of p. p = ................................................. [4]

Mark scheme: 6(a) 4.5 oe 2 B1 for 7 x  3x = 13  5 oe 6(b) 15 3 B1 for 8 x  12 3 6 x oe oe 14 M1 for correctly collecting terms in an equation 6(c) 2 3 B2 for 3 x  2 4 oe ,  2 oe 3 or for 3 3 x  2  x  2   0  oe 4 4(3)( 4) or for oe 2(3) or M1 for  3 x  2  2  8  2 oe 6(d) 0.35 –2.85 3 B2 for –2.851 to –2.850 and 0.350 to 0.351 OR M2 for correct sketch indicating both roots 5 5 2  4(2)(  2) or for 2(2) or M1 for 2 x 2  5 x  2   0  or  2 x 2  5 x  2   0  6(e)  x  10 y 4 3 M1 for logy4 B1 for 1 = log10 6(f) 1 4 B3 for n = 3 oe 4 n n  6 or M2 for 212  18 or for 12p = 36p – 6 oe n n  6 or M1 for p  or 2 p  12 18 n n  6 or for and seen 12 18

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7 y 2 – 90 0 90 x – 2 f ( x) = 2 cos ( x - 45)° - 1 for values of x between - 90 and 90. (a) On the diagram, sketch the graph of y = f ( x) . [3] (b) Write down the x‑coordinates of the points where the curve meets the x‑axis. x = .................. or x = .................. [2] (c) Write down the coordinates of the local maximum point. ( ...................... , ...................... ) [1] (d) The line y = 0.005 x intersects the curve y = 2 cos ( x - 45 )° - 1 three times. (i) Find the x‑coordinates of the points of intersection. x = .................. or x = .................. or x = .................. [3] (ii) Solve the inequality. 2 cos ( x - 45)° - 1 2 0.005x ................................................................................ [2]

Mark scheme: y f(x)=abs(2cos(x-0.785))-1 1.8 7(a) 1.6 3 Correct graph 1.4 1.2 1 0.8 B1 for 1 max in first quadrant 0.6 0.4 0.2 x B1 for V shape at approx x = –45 -1.7 -1.6 -1.5 -1.4 -1.3 -1.2 -1.1 -1 -0.9 -0.8 -0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 -0.2 -0.4 -0.6 -0.8 -1 -1.2 -1.4 -1.6 -1.8 7(b) –75, 2 B1 for each –15 7(c) (45, 1) 1 7(d)(i) –64.8 or –64.76... 3 B1 for each –17.9 or – 17.92... or for –65, –18, 89 88.8 or 88.78.... If 0 scored SC1 for sketch of y = 0.005x 7(d)(ii) x < –64.8 2 FT –17.9 < x < 88.8 B1 for one correct

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8 (a) Solve the simultaneous equations. 5x - 4y = 13 3x + 2y =- 1 You must show all your working. x = ................................................. y = ................................................. [3] 1 (b) f ( x) = 3 x + 1 g ( x) = , x ! 1.5 2x - 3 (i) Find f ( - 2) . ................................................. [1] (ii) Find f ( f ( x)) , giving your answer in its simplest form. ................................................. [2] 1 (iii) Solve g ( f ( x)) = . 5 x = ................................................. [3]

Mark scheme: 8(a) Correct method to eliminate one variable M1 [ x  ] 1 A2 If 0 scored, SC1 for answers that [ y ] 2 satisfy one equation 8(b)(i) –5 1 8(b)(ii) 9x + 4 final answer M1 for 3  3 x  1  1 oe 2 8(b)(iii) 1 3 M2 for 2  3 x  1  3  5 oe 1 1 or M1 for  oe 2  3 x  1  3 5 OR M2 for f(x) = 4 oe 1 1 or M1 for  2f x  3 5

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9 There are 80 students in a school year, 44 boys and 36 girls. Each student chooses their favourite sport. The number of boys and the number of girls choosing each sport is shown in the table. Athletics Football Hockey Swimming Boys 12 16 8 8 Girls 5 3 17 11 (a) A student is chosen at random from the 80 students. Find the probability that the student chosen is (i) a girl whose favourite sport is athletics ................................................. [1] (ii) a boy whose favourite sport is not football. ................................................. [1] (b) One of the girls is chosen at random. Find the probability that her favourite sport is hockey. ................................................. [2] (c) Three of the boys are chosen at random. (i) Find the probability that one of the boys chooses athletics, one of them chooses football and the other chooses swimming. ................................................. [1] (ii) Calculate the probability that the three boys each have a different favourite sport. ................................................. [3]

Mark scheme: 9(a)(i) 5 1 oe 80 9(a)(ii) 28 1 oe 80 9(b) 17 2 k 17 oe M1 for where k < 36 or where 36 36 m 17  m  80 9(c)(i)  12 16 8  384 1    6  oe    44 43 42  3311 9(c)(ii) 3648 1216 3 M2 for or oe 9933 3311  12 16 8 12 16 8         44 43 42 44 43 42   k 6  or 0.367 or 0.3672 to 0.3673  12 8 8 16 8 8           44 43 42 44 43 42  or M1 for any two of these products seen

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10 B 78° NOT TO SCALE O A C 20° E D A, B, C and D lie on a circle, centre O. DE is a tangent to the circle at D. ACE is a straight line. Find (a) angle AOC Angle AOC = ................................................. [1] (b) angle OAC Angle OAC = ................................................. [2] (c) angle ADC Angle ADC = ................................................. [1] (d) angle CAD. Angle CAD = ................................................. [3]

Mark scheme: 10(a) 156 1 10(b) 12 2 180  their ( a ) M1 for 2 10(c) 102 1 10(d) 29 3 M2 for x x their102  20  180 oe or M1 for indicating angle CDE = angle DAC

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11 E 16 cm C D NOT TO SCALE 34 cm A 20 cm B The diagram shows a cuboid with base ABCD. AB = 20 cm , BC = 34 cm and CE = 16 cm . Water is poured into the cuboid to a height of 8 cm. (a) Find the volume of water in the cuboid. .......................................... cm3 [2] (b) A sphere of radius 4 cm is placed so that it rests on the base of the cuboid. The water level is now q cm above the base of the cuboid. Find the value of q. ................................................. [4] (c) The sphere is removed from the cuboid. 15 identical cubes of side x cm are placed so that they rest on the base of the cuboid. (i) Find the maximum value of x. x = ................................................. [3] (ii) The water level is now p cm above the base of the cuboid. Find the maximum value of p. p = ................................................. [3]

Mark scheme: 11(a) 5440 2 M1 for 20  34  8 or 20  34  16 11(b) 8.39 or 8.394… 4 B3 for 0.394 or 0.394… OR 4 3  4  their a  3 () M3 for q  680 or M2 for 4 3  20  34  q   4 their ( a )  3 4 3 or M1 for  4 3 11(c)(i) 6.67 or 6.666 to 6.667 3 20 34 M2 for and 3 5 20 34 or M1 for or 3 5 or recognition of 3 by 5 11(c)(ii) 14.5 or 14.53 to 14.55 3 B2 for 6.53 to 6.55 OR M2 for 3 15   their (c)(i)  their (a) p  20  34 or M1 for 3 20  34 p  15   their (c)(i)    their (a) 

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12 A B D 30° NOT TO SCALE 200 cm C The diagram shows two right‑angled triangles ABC and CBD. AB = BC , AC = 200 cm and angle BDC = 30° . Find the perimeter of triangle ACD. ............................................ cm [6]

Mark scheme: 12 61.5 or 61.46…. 6 B1 for AB or BC = 10 their BC M2 for CD  oe sin 30 their BC or M1 for sin30  oe CD their BC M2 for BD  oe tan 30 their BC or M1 for tan30  oe BD

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