Cambridge IGCSE Mathematics - International 0607 — 2017 May/June Paper 4 · Variant 2

1 In 2016, Carla’s salary was $23 970 per year. (a) From her salary she pays tax at a rate of 20%. She is paid monthly in equal amounts. Calculate the amount Carla receives each month after tax has been paid. $ .................................................... [3] (b) Carla’s salary of $23 970 was 2% more than her salary in 2015. (i) Calculate her yearly salary in 2015. $ .................................................... [3] (ii) From 2016, Carla’s employer agrees to pay her an increase of 3% each year. Calculate the year in which her salary is first greater than $30 000. .................................................... [3]

Mark scheme: Question Answer Marks Partial Marks 1(a) 1598 final answer 3 M2 for (23 970 × 0.8) ÷ 12 oe or M1 for 23 970 × 0.2 or better or for 23 970 ÷ 12 1(b)(i) 23 500 nfww 3 M2 for 23 970 ÷ 1.02 oe or M1 for 23 970 = 102% 1(b)(ii) 2024 nfww 3  30000  log    23970  M2 for oe soi by 7.59...or sketch log1.03 leading to 7.59 or 2 trials, one giving 7 and one giving 8  3  n or M1 for 23 970 ×  1 +  [= 30 000] oe seen  100  or reasonable sketch or 3 trials of 23 970 × 1.03n or 1 trial giving 8.

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2 (a) (i) Reflection in the line y = x maps triangle A onto triangle B. Describe fully the single transformation that maps triangle B onto triangle A. ........................................................................................................................................................... ..................................................................................................................................................... [1] (ii) Enlargement, with centre (2, 1) and scale factor 4, maps triangle C onto triangle D. Describe fully the single transformation that maps triangle D onto triangle C. ........................................................................................................................................................... ..................................................................................................................................................... [2] J N - 3 (iii) Translation by the vector KK OO maps triangle E onto triangle F. 5 L P Describe fully the single transformation that maps triangle F onto triangle E. ........................................................................................................................................................... ..................................................................................................................................................... [2] (b) y 4 3 2 1 P x 0 –7 –6 –5 –4 –3 –2 –1 1 2 3 4 5 6 7 –1 –2 –3 –4 (i) Rotate triangle P through 90° anticlockwise about (0, 0). Label the image Q. [2] (ii) Stretch triangle P with stretch factor 2 and the y-axis invariant. Label the image R. [2]

Mark scheme: 2(a)(i) Reflection, y = x 1 2(a)(ii) Enlargement [with centre] (2, 1) 2 B1 for each 1 [scale factor] oe 4 2(a)(iii) Translation 2 B1 for each  3     −5  2(b)(i) Correct triangle 2 SC1 for rotation 90° clockwise about (0, 0) (0, 0), (0, 2), (–2, 3) or rotation 90° anti-clockwise about different centre 2(b)(ii) Correct triangle 2 SC1 for stretch with s.f. = 2, x-axis invariant or (0, 0), (4, 0), (6, 2) stretch with y-axis invariant with different scale factor.

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3 Two judges each give a mark out of ten for each dancer in a competition. Their marks for ten dancers are shown in the table. Mark from 4.0 4.6 5.2 6.2 8.8 6.8 7.0 7.4 8.0 8.6 judge A (x) Mark from 3.8 4.0 4.4 5.0 7.6 5.2 5.6 6.8 6.6 7.0 judge B (y) (a) Complete the scatter diagram. The first four points have been plotted for you. y 10 9 8 7 6 Mark from judge B 5 4 3 2 1 x 0 1 2 3 4 5 6 7 8 9 10 Mark from judge A [3] (b) What type of correlation is shown on your scatter diagram? .................................................... [1] (c) (i) Find the equation of the regression line, in the form y = mx + c. y = .................................................... [2] (ii) Judge A gives another dancer a mark of 6.4 . Use your equation to estimate the mark judge B gives this dancer. .................................................... [1]

Mark scheme: 3(a) 6 points correct 3 B2 for 4 or 5 correct or B1 for 2 or 3 correct 3(b) Positive 1 3(c)(i) y = 0.787x + 0.356 final answer 2 0.7874 to 0.7875, 0.3555 to 0.3556 B1 for one correct or for y = 0.79x + 0.36 final answer 3(c)(ii) 5.4[0] 1 FT from their (c)(i)

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=4 p = KK OO and q KK OO 2 3 L P L P (a) Find 1 (i) the column vector p , 2 J N K O K O [1] K O L P (ii) the column vector q − 2p, J N K O K O [2] K O L P (iii) p , leaving your answer in surd form. .................................................... [2] (b) AB = p + q Mark and label point B on the grid. A [2]

Mark scheme: 4(a)(i)  −1.5  1   oe  1  4(a)(ii)  10  2 B1 for each    − 1  4(a)(iii) 13 final answer 2 M1 for (–3)2 + 22 oe soi by 3.61 or 3.605 to 3.606 13 in working implies M1 4(b) Correct B clearly indicated 2 1 1 B1 for vector  drawn not from A or  seen 5 5 or correctly following through, from A, their incorrect vector seen.  −3  4 or either   or  correctly drawn only if one  2  3 starts from A.

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5 Nitini flies from New Delhi to Singapore for a holiday. (a) Nitini changes 119 050 Indian rupees (INR) to Singapore dollars (SGD). The exchange rate is 1 SGD = 47.62 INR. Find how many Singapore dollars he receives. .......................................... SGD [2] (b) The flight from New Delhi to Singapore takes 5 hours and 45 minutes. The distance of the flight is 4150 km. (i) The time in New Delhi when the flight leaves is 21 55. The time in Singapore is 2 12 hours ahead of the time in New Delhi. Find the time in Singapore when the flight arrives. .................................................... [2] (ii) Find the average speed of the aircraft. .......................................... km / h [3] (iii) On the return flight the average speed is 750 km / h. Find the time of this flight in hours and minutes. .................. h ....................... min [3]

Mark scheme: 5(a) 2500 2 M1 for 119 050 ÷ 47.62 5(b)(i) [0]6 10 or 6 10 am oe 2 B1 for [0]0 25 or [0]3 40 or 28 h 130 min oe seen 5(b)(ii) 722 or 721.7 ... 3 M1 for 4150 ÷ their 5h 45 min B1 for 5.75 oe 5(b)(iii) 5 h 32 (or 31.8 to 32[.0] ) min 3 M1 for 4150 ÷ 750 soi by 5.53 or 5.53 ... B1FT for correct conversion to hours and minutes

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6 (a) (i) x is proportional to v. Write down an expression for x in terms of v and a constant c. x = .............................................. [1] (ii) y is proportional to v2. Write down an expression for y in terms of v and a constant k. y = .............................................. [1] (iii) d = x + y Write down an expression for d in terms of v, c and k. d = .............................................. [1] (b) The table shows two values of v and the corresponding values of d. v d 12 750 20 2050 Using your answer to part (a)(iii), (i) show that 125 = 2c + 24k, [1] (ii) write down a second equation connecting c and k. .................................................... [1] (c) Solve the simultaneous equations in part (b) to find the value of c and the value of k. c = .................................................... k = .................................................... [3] (d) Find the value of d when v = 40. d = .................................................... [2]

Mark scheme: 6(a)(i) [x =] cv oe 1 6(a)(ii) [y =] kv2 oe 1 6(a)(iii) [d =] cv + kv2 or v(c + kv) oe 1 FT 6(b)(i) 750 = 12c + 122 k oe M1 isw any cancelling 6(b)(ii) 2050 = 20c + 202 k oe 1 isw any cancelling 6(c) [c =] 2.5 oe cao 3 M1 for correctly eliminating one variable from [k =] 5 cao their equations in this part. or sketches of lines A1 for either solution If zero scored SC1 for their values satisfying one equation. 6(d) 8100 2 M1 for correct substitution of 40 into their (a)(iii) containing their values of c and k.

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7 A ship sails 65 km on a bearing of 310° from A to B. It then changes course and sails 40 km on a bearing of 250° from B to C. The ship then returns to A. (a) On the diagram, sketch the path of the ship from A. On your diagram show the bearings and distances. North A [3] (b) Find angle ABC. .................................................... [1] (c) Calculate AC and show that it rounds to 91.8 km, correct to the nearest tenth of a kilometre. [3] (d) Find the bearing of C from A. .................................................... [4]

Mark scheme: 7(a) Correct skettch showing bearingsb 3 B1B for 310° bearingb apprrox correct (2270 to 360) aand and distancees markedm B1B for 250° bearingb apprrox correct (1180 to 270) aand markedm B1B for distannces correctlyy marked 7(b) 120 1 7(c) 402 + 652 – 22×40×65×coos their 120 M1 theirt 120 muust be betweeen 0 and 180 40 2 + 655 2 − [ ]2 AllowA cos1220 = 2 × 400 × 65 91.78 to 91.779 A2 A1A for 8425 or 5 337 7(d) 288 or 287.88... 4 40sin ( their120 ) M2M for oe 91.8 sinθ sin(thheir 120) oro M1 for = oe 404 991.8 IfI cosine rulee used, M2 fofor explicit exxpression or M1M for impliicit. A1A for 22.2 oro 22.16 to 222.17… IfI 0 scored SC2 for answwer 108 or 1007.8…

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8 y 5 x –400 0 600 f(x) = 3sinx (a) Sketch the graph of y = f(x) for - 400 ° G x G 600 ° . [3] (b) Find the x co-ordinates of the local maximum points of f(x) for - 400° G x G 600° . x = .......................... or x = .......................... or x = .......................... [3] (c) The point (30, 3) is on the graph. The point (a, 3) is also on the graph where 600° 1 a 1 900 ° . Find the two possible values of a. a = .......................... or a = .......................... [2] x (d) g(x) = 3 − 100 Solve the inequality g x 2 f x . ^ h ^ h ............................................................................................................................................................. [3]

Mark scheme: 8(a) 3 WithW correct shape with ttwo max on right of y-axxis anda one on leeft, all abovee x-axis and reasonabler qualityq oro B2 for corrrect shape anand all above x-axis oro B1 for corrrect shape Correct skettch 8(b) –270, 90, 4550 3 B1B for each SC2S for all correctc but wwith y co-ordss oro SC1 for twwo correct wwith y co-ordss 8(c) 750, 870 2 B1B for each 8(d) x < 54.7 1 54.745 to 54.775 164 < x < 2667 2 163.51 to 163.6 , 266.6... B1B for one innequality oro B1 for botth values seeen IfI 0 scored, B1B for straighht line with negativen gradientg crosssing curve thhree times beetween xx = 0 and x == 400. May bbe freehand.

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9 C x cm NOT TO SCALE 60° A B (x + 2) cm In the diagram AC = x cm, AB = (x + 2) cm and angle A = 60°. (a) (i) Find an expression, in terms of x, for the area of triangle ABC. Give your answer in surd form. .............................................cm2 [2] (ii) The area of triangle ABC = 18 3 cm2. Show that x2 + 2x – 72 = 0. [2] (b) (i) Solve the equation x2 + 2x – 72 = 0. x = .......................... or x = .......................... [2] (ii) Find the shortest distance between the line AB and the point C. .............................................. cm [2]

Mark scheme: 9(a)(i) 1 3 2 1 x + 2 ) × sin60 × x × ( x + 2 ) × oe or better M1 for × x × ( 2 2 2 final answer 9(a)(ii) equating to 18 3 and correct M1 Dependent on correct answer used from (a)(i) or answer to (a)(i) contains sin60 but is elimination of 3 otherwise correct. Completion with at least one step A1 No errors or omissions 9(b)(i) 7.54 or 7.544... , –9.54 or –9.544... 2 B1 for each If 0 scored, M1 for substitution in formula or sketch or (x + 1)2 – 73 or better 9(b)(ii) 6.53 or 6.54 or 6.529 to 6.536... 2 [ ] M1 for sin 60 = oe their 7.54

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10 y C (14, 8) NOT TO SCALE B (11, 4) A (2, 2) x 0 A is the point (2, 2), B is the point (11, 4) and C is the point (14, 8). (a) Find the equation, in the form y = mx + c, of (i) the line AC, y = .................................................... [3] (ii) the line through B that is perpendicular to AC. y = .................................................... [3] (b) Show that the point (10, 6) is on both the lines you found in part (a). [2] (c) AC is the perpendicular bisector of BD. Find the co-ordinates of D. (...................... , .......................) [1] (d) Find the exact area of the quadrilateral ABCD. .................................................... [4]

Mark scheme: 10(a)(i) 1 3 8 − 2 [y =] x + 1 M1 for gradient = oe 2 14 − 2 M1 for correct substitution of (2, 2) or (14, 8) into y = (their m)x + c oe soi 10(a)(ii) [y =] –2x + 26 3 −1 M1 for gradient = their 12 M1for substituting (11, 4) into y = (their – 2 )x + c oe soi 10(b) Correct substitution and completion 2 B1 for either of (10, 6) for both lines oe OR M1 for correct elimination of x or y from equations A1 for completion to solution (10, 6) 10(c) (9, 8) 1 10(d) 30 cao 4 1 2 2 2 2 M3 for × 12 + 6 × 2 + 4 oe  2 or B2 for two of 12 2 + 6 2 oe (AC), 2 2 + 4 2 oe (BD or MC), 8 2 + 4 2 oe (AM), 2 2 + 12 oe (MD or MB) or B1 for one of these. (M is the intersection of AC and BD) OR M3 for full area e.g. [0.5 × 12 × 6 – 0.5 × 6 × 7] × 2 or B2 for 2 correct areas evaluated or B1 for 1 correct area evaluated

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11 A farmer sorts the grapefruit he grows into sizes, according to their diameter. The diameters, d cm, of 170 grapefruit are shown in the table. Size Small Medium Large Very Large Diameter (d cm) 9 1 d G 10 10 1 d G 12 12 1 d G 14 14 1 d G 17 Frequency 10 50 65 45 (a) Calculate an estimate of the mean diameter of the grapefruit. .............................................. cm [2] (b) On the grid, draw a histogram to represent this information. Complete the scale on the frequency density axis. Frequency density d 8 9 10 11 12 13 14 15 16 17 18 Diameter (cm) [4] (c) Two of the 170 grapefruit are chosen at random. Calculate the probability that (i) they are both Very Large, .................................................... [2] (ii) one is Small and the other is Medium. .................................................... [3]

Mark scheme: 11(a) 12.9 or 12.86 to 12.87 2 M1 for evidence of at least three mid-interval values 9.5, 11, 13, 15.5 soi by 95, 550, 845, 697.5 or 2187.5 11(b) Correct Histogram 4 B1 for correct bar widths no gaps B3 for 4 correct heights and corresponding scale from 0 or B2 for 3 correct heights and corresponding scale from 0 or B1 for 2 correct heights and corresponding scale from 0 or B1 for 3 correct frequency densities soi 11(c)(i) 198 2 45 44 oe M1 for × 2873 170 169 11(c)(ii) 100 3 10 50 50 10 oe M2 for × + × oe 2873 170 169 170 169 10 50 or M1 for × 170 169

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12 f(x) = 4x + 2 g(x) = 5 − 2x h(x) = x2 − 3 (a) Find g(−3). .................................................... [1] (b) Find f(h(2)). .................................................... [2] (c) Find x when f(x) = −10. x = .................................................... [2] (d) Write down the range of h(x). .................................................... [1] (e) Find f−1(x). f−1(x) = .................................................... [2] (f) k(x) = 10 − 4x Describe fully the single transformation that maps the graph of y = g(x) onto the graph of y = k(x). ................................................................................................................................................................... ............................................................................................................................................................. [3] J N 2 (g) The graph of y = h(x) is translated by the vector KK OO . 0 L P Find the equation of the graph of the image. Write your answer in the form y = ax2 + bx + c. y = .................................................... [3]

Mark scheme: 12(a) 11 1 12(b) 6 2 B1 for h(2) = 1 soi or 4(x2 – 3) + 2 or better 12(c) –3 2 M1 for 4x = –10 – 2 12(d) h(x) ⩾ –3 1 Allow y ⩾ –3 12(e) x − 2 2 y 2 oe final answer M1 for y – 2 = 4x or x = 4y + 2 or = x + 4 4 4 12(f) Stretch 3 B1 for each x-axis invariant [Scale factor] 2 OR Reflection M1 y = −2.70 + 6.75 A2 OR Rotation M1 (2.5, 0) A1 167 (167.47) or 12.5 (12.53) A1 clockwise 12(g) [y =] x² – 4x + 1 3 M2 for y = (x – 2)² – 3 or M1 for x – 2 seen in a quadratic If 0 scored, SC1 for y = x² + 4x + 1

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