Cambridge IGCSE Mathematics - International 0607 — 2024 Feb/March Paper 4 · Variant 2

1 y 1.5 x -180 0 180 -1.5 f ( x) = ( sin x°) 2 (a) On the diagram, sketch the graph of y = f ( x) for - 180 G x G 180 . [2] (b) Write down the amplitude and period of f(x). Amplitude ................................................ Period ................................................ [2] (c) g ( x) = 0 .002 x + 0 .5 (i) On the diagram, sketch the graph of y = g ( x) for - 180 G x G 180 . [2] (ii) Solve g ( x) = f ( x) for - 180 G x G 180 . ............................................................................................................................................. [4] (iii) Solve g ( x) 1 f ( x) for - 180 G x G 180 . ............................................................................................................................................. [2]

Mark scheme: Question Answer Marks Partial Marks 1(a) Correct curve 2 B1 if cusp at (0, 0) or ‘correct curve’ but height clearly incorrect. 1(b) 0.5 2 B1 for each 180 1(c)(i) Correct sketch 2 maximum of 1 mark if it does not intersect curve 4 times B1 for positive gradient and positive y- intercept 1(c)(ii) –154 or –154.0... 4 B1 for each –40.4 or –40.36 to –40.35 Max 3 if y coordinates included. 50.9 or 50.87... 121 or 120.5 to 120.6 1(c)(iii) –154 < x < –40.4 2 FT from (c)(ii) 50.9 < x < 121 B1 for each Same accuracy as (ii)

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2 Asif, Basheera and Chelsea make baskets. (a) The selling price of a basket increases by 8%. The new selling price is $4.86 . Find the original selling price of a basket. $ ................................................ [2] (b) Asif earns $4.70 per hour plus $1.21 for each basket he makes. Each week he works 8 hours a day for 5 days. Each day Asif makes 18 baskets. Calculate the total amount Asif earns in one week. $ ................................................ [3] (c) One day Basheera and Chelsea make a total of 36 baskets. They each work for 8 hours. Basheera takes x minutes to make a basket. Basheera takes 6 minutes longer than Chelsea to make a basket. (i) Write down an expression in terms of x for the number of baskets Chelsea makes. ................................................. [1] (ii) Write down an equation in terms of x and show that it simplifies to 3x 2 - 98x + 240 = 0 . [3] (iii) Solve the equation 3x 2 - 98 x + 240 = 0 . x = ....................... or ....................... [2] (iv) Find the number of baskets Chelsea makes. ................................................. [2]

Mark scheme: 2(a) 4.5[0] 2  8  M1 for A   1 +  = 4.86 oe  100  2(b) 296.9[0] 3 M1 for 8 [× 5] × 4.7[0] oe M1 for [5 ×] 18 × 1.21 oe 2(c)(i) 480 1 oe x − 6 2(c)(ii) 480 480 M1 480 + = 36 oe Allow + their (i) = 36 x x − 6 x 480(x – 6) + 480x = 36x(x – 6) oe M1 Clearing fractions correctly (Dep on two fractions with different algebraic denominators) Completion to A1 3x2 – 98x + 240 [= 0] with no errors 2(c)(iii) 8 2 M1 for suitable sketch of parabola with 2 30, oe positive solutions 3 or (3x – 8)(x – 30) = 0 −−( 98)  ( −98) 2 − 4(3)(240) or 2  3 2(c)(iv) 20 2 480 M1 for their 30 − 6

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3 y 8 7 6 5 B 4 C 3 A 2 1 0 x -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 -1 -2 -3 -4 -5 -6 -7 -8 (a) Describe fully the single transformation that maps triangle A onto triangle B. ..................................................................................................................................................... ..................................................................................................................................................... [2] (b) Describe fully the single transformation that maps triangle A onto triangle C. ..................................................................................................................................................... ..................................................................................................................................................... [3] (c) (i) Rotate triangle A through 90° clockwise about ( - 1 , - 1) . Label the image D. [2] (ii) Reflect triangle D in the line x = - 1. Label the image E. [2] (iii) Describe fully the single transformation that maps triangle A onto triangle E. ............................................................................................................................................. ............................................................................................................................................. [2]

Mark scheme: 3(a) Translation 2 B1 for each  −7     2  3(b) Enlargement 3 B1 for each 1 [sf] − 2 [centre] (–1, 3) 3(c)(i) Triangle at (1, –3), (5, –3), (1, –5) 2 B1 for rotation 90˚ anticlockwise about (–1, –1) or for rotation 90˚ clockwise about wrong centre 3(c)(ii) Triangle at (–3, –3), (–7, –3), 2 FT their (c)(i) (–3, –5) B1 for reflection in x = k or y = –1 3(c)(iii) Reflection 2 B1 for each y = –x – 2 oe

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4 Ten people trained for a fitness test. The table shows the amount of time they each trained and the time they each took to do the test. Training time 12.2 9.3 16.4 7.2 15.6 13.7 9.4 13.1 12.8 14.2 (x hours) Time for test 6.5 7.4 5.3 8.1 5.1 6.3 7.6 6.6 6.9 5.7 (y minutes) (a) Complete the scatter diagram. The first 6 points have been plotted for you. y 9 8 7 Time for test (min) 6 5 4 3 x 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Training time (hours) [2] (b) What type of correlation is shown on the scatter diagram? ................................................. [1] (c) Find the equation of the regression line. Give your answer in the form y = mx + c . y = ................................................ [2] (d) Anna trained for 10.8 hours. Use your equation to estimate the time Anna took for the test. .......................................... min [1] (e) Ben trained for 20.5 hours. Explain why you should not use your equation to estimate the time Ben took for the test. ..................................................................................................................................................... [1]

Mark scheme: 4(a) 4 points plotted correctly 2 B1 for 2 correct. 4(b) Negative 1 4(c) y = – 0.325x + 10.6 2 0.3245... , 10.57... B1 for y = – 0.325x + k or y = kx + 10.6 4(d) 7.06 to 7.09 1 FT their (c) 4(e) Too far outside range of data oe 1

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5 B North NOT TO SCALE 123 m 154 m North A 27° 183 m C 106° D The diagram shows a field ABCD, with a straight path AC. The bearing of C from A is 122° . (a) Calculate the bearing of D from C. ................................................. [3] (b) Show that angle ABC = 81.9° correct to one decimal place. [3] (c) Find the total area of the field ABCD. ........................................... m2 [5]

Mark scheme: 5(a) 255 cao 3 B2 for 75 correctly referenced at C or D OR B1 for angle ACD = 47 B1 for angle ACN(orth) = 58 or ACS(outh) = 122 5(b) 1232 + 154 2 − 1832 M2 M1 for 1832 = 1232 + 1542 – 2 × 123 × [cos] = 154 × cos[…] 2  123  154 81.87... A1 5(c) 15 200 or 15 150 to 15 161 5 183sin27 M2 for CD = sin106 CD 183 or M1 for = sin27 sin106 OR 183sin their 47 M2 for AD = sin106 AD 183 or M1 for = sin their 47 sin106 AND M1 for 1 × 154 × 123 × sin 81.9 2 M1 for 1 × 183 × their CD × sin their47 2 or 1 × 183 × their AD × sin 27 2 or 12 × their CD × their AD × sin 106

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6 (a) U = {integers from 1 to 15} P = {factors of 12} Q = {multiples of 3} (i) Complete the Venn diagram. U P Q [2] (ii) Write down the elements of P + Q . ................................................. [1] (iii) Find n ( P l + Q ) , ( P + Q l ) _ i. ................................................. [1] (b) Bag A Bag B Bag A contains 4 black balls and 3 white balls. Bag B contains 2 black balls and 4 white balls. (i) Amy picks a ball at random from bag A. She notes the colour of the ball and replaces it in bag A. Find the probability that Amy’s ball is black. ................................................. [1] (ii) Basma picks two balls at random from bag B. She notes the colour of each ball and replaces them in bag B. Find the probability that both balls are white. ................................................. [2] (iii) Basma chooses one bag at random. She picks one ball at random from this bag. Find the probability that the ball is white. ................................................. [3]

Mark scheme: 6(a)(i) 2 B1 for 1 or 2 elements misplaced or omitted. 1 2 3 9 6 4 12 15 5 7 8 10 11 13 14 6(a)(ii) 3, 6, 12 1 FT their Venn diagram 6(a)(iii) 5 1 FT their Venn diagram 6(b)(i) 4 7 6(b)(ii) 2 2 4 3 oe M1 for  oe 5 6 5 6(b)(iii) 23 3 1 3 1 4 oe M2 for  +  42 2 7 2 6 or M1 for one of above products

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7 (a) F 30° E NOT TO SCALE G 130° A B C D ABCD is a straight line and EC and BF meet at G. BE is parallel to CF and GF = CF . Angle ABE = 130° and angle BFC = 30° . Find (i) angle FCD Angle FCD = ................................................ [2] (ii) angle FBC Angle FBC = ................................................ [1] (iii) angle BGE. Angle BGE = ................................................ [2] (b) B NOT TO SCALE C X A D A, B, C and D are points on the circle. AC and BD meet at X. (i) Show that triangles AXB and DXC are similar. Give a reason for each statement you make. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. ............................................................................................................................................. [2] (ii) AX = 5 cm, XC = 2 cm and XD = 4 cm. Find the length of BD. BD = ........................................... cm [3]

Mark scheme: 7(a)(i) 50 2 B1 for angle BCF = 130 or angle EBC = 50 soi by angle EBG = 30 and angle GBC = 20 or angle FCG = 75 and angle GCB = 55 7(a)(ii) 20 1 7(a)(iii) 75 2 180 − 30 M1 for 2 7(b)(i) 2 from 2 B1 for 2 pairs correct with no/incorrect Angle AXB = Angle DXC reasons and conclusion [Vertically] opposite angles or for one pair correct with reason. Angle ABX = Angle DCX Angles in same segment. Angle BAX = Angle CDX Angles in same segment. And conclusion AA[A] 7(b)(ii) 6.5 3 B2 for BX = 2.5 ... 5 or M1 for = oe 2 4

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8 The table shows the money received in a shop for 120 days. Money received Frequency ($x) 500 1 x G 1000 6 1000 1 x G 1500 16 1500 1 x G 2000 24 2000 1 x G 2500 36 2500 1 x G 3000 20 3000 1 x G 3500 14 3500 1 x G 4000 4 (a) On the grid, draw a cumulative frequency curve to show this information. 120 100 80 Cumulative frequency 60 40 20 0 x 500 1000 1500 2000 2500 3000 3500 4000 Money received ($) [4] (b) Use your curve to estimate (i) the median $ ................................................ [1] (ii) the interquartile range. $ ................................................ [2] (c) Use your curve to estimate the percentage of these 120 days where the shop received more than $1800. ............................................. % [3]

Mark scheme: 8(a) Correct graph through 7 points 4 B3 for graph through 5 correct points or B2 for graph through 3 correct points or for all correct heights translated to other point in interval or for correct points plotted not joined or B1 for 6, 22, 46, 82, 102, 116, 120 8(b)(i) 2100 to 2300 1 FT from increasing curve. 8(b)(ii) 950 to 1150 2 B1 for 1650 ⩽ [LQ] ⩽ 1750 or 2600 ⩽ [UQ] < 2750 8(c) 68.3 to 71.7 3 B1FT for 34 to 38 their M1 for (34to38)[100] 120 120 − their or (34to38)[100] 120

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9 y A NOT TO SCALE B O x A is the point (-4, 6) and B is the point (8, 2). (a) Find the coordinates of the mid-point of AB. (..................... , .....................) [2] (b) Find the equation of AB. ................................................. [3] (c) Show that the equation of the perpendicular bisector of AB is y = 3x - 2 . [3] (d) The point C has coordinates (3, 7). Show that C lies on the perpendicular bisector of AB. [1] (e) Find the area of triangle ABC. ................................................. [4]

Mark scheme: 9(a) (2, 4) 2 B1 for each coordinate 9(b) 1 2 3 2 − 6 y = − x + 4 oe cao M1 for 3 3 8 −−( 4) final answer M1 for substituting (2, 8) or (–4, 6) into  1  y = their  −  x + c oe  3  9(c)  1  M1 Gradient = for –1 ÷  their −  oe  3  substituting their (2, 4) into M1 y = their 3 x + c oe Completion to y = 3x – 2 with no errors A1 Dep on M1, M1 or omissions 9(d) 3 × 3 – 2 = 7 1 9(e) 20 4 2 2 M1 for [AB =] ( 8 + 4 ) + ( 2 − 6 ) M1 for [h =] ( 7 − their 4 ) 2 + ( 3 − their 2 ) 2 1 M1 for  their 160  their 10 2

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10 f ( x) = 3 x - 2 g ( x) = 5 - 2 x h ( x) = x2 (a) (i) Find g ( - 2) . ................................................. [1] (ii) Find h ( g ( x)) . Write your answer in the form ax 2 + bx + c . ................................................. [3] (iii) Find g -1 ( x) . g -1 ( x) = ................................................ [2] f ( x) (b) (i) On the diagram, sketch the graph of y = for values of x between -2 and 6 . g ( x) y 10 0 x -2 6 -10 [3] f ( x) (ii) An asymptote to the graph of y = is parallel to the y-axis. g ( x) Find the equation of this asymptote. ................................................. [1] f (x) (iii) Solve = 5 - 2x . g (x) ................................................ [3]

Mark scheme: 10(a)(i) 9 1 10(a)(ii) 4x2 – 20x + 25 final answer 3 M1 for (5 – 2x)2 B1 for 3 terms correct in 25 – 10x – 10x + 4x2 10(a)(iii) 5 −x 2 y 5 oe final answer M1 for x = 5 – 2y or = − x 2 2 2 or y – 5 = –2x 10(b)(i) Correct graph 3 B2 for correct but some ‘curl back’ or overlap or too wide a gap. B1 for one branch correct. 10(b)(ii) x = 2.5 oe 1 10(b)(iii) 1.67 or 1.670 to 1.671 3 B2 for one solution 3.31 or 3.307 to 3.308 Max 1 if y coordinates included. or M1 for sketch of y = 5 – 2x If 0 scored, SC1 for only y values seen in answer space, with correct x values seen in working.

More questions on Functions

11 (a) Solve. 3x + 2 2 7x - 8 ................................................. [2] (b) Factorise fully. 75x 2 - 3 ................................................. [2] (c) Simplify. 2 1 1 (i) + - 3x 6x 5x ................................................. [2] 2 x 2 + 3 x - 2bx - 3b (ii) 2 2x - 7x - 15 ................................................. [4]

Mark scheme: 11(a) x < 2.5 oe final answer 2 M1 for 2 + 8 > 7x – 3x oe or B1 for x * 2.5 where * is =, >, ≤ or ≥ 11(b) 3(5x + 1)(5x – 1) final answer 2 B1 for 3(25x2 – 1) or (15x + 3)(5x – 1) or (15x – 3)(5x + 1) 11(c)(i) 19 2 B1 for any equivalent cao final answer or M1 for correct use of common 30x denominator 20 + 5 − 6 20 x + 5 x − 6 x e.g. , etc. oe 30 x 30 x 2 11(c)(ii) x − b 4 B3 for (x – b)(2x + 3) and (x – 5)(2x + 3) final answer or B2 for (x – b)(2x + 3) x − 5 or for (x – 5)(2x + 3) or B1 for x(2x + 3) – b(2x + 3) or 2x(x – b) + 3(x – b) or x(2x + 3) – 5 (2x + 3) or 2x(x – 5) + 3 (x – 5) or (2x + c)(x + d) where c + 2d = –7 or cd = –15

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