Cambridge IGCSE Mathematics - International 0607 — 2017 Oct/Nov Paper 4 · Variant 3

1 f (x) = 3x - 2 g (x) = , x =Y 0 h ( x) = (x + 2) 2 x (a) Find (i) f(4), .................................................... [1] (ii) gf(4). .................................................... [1] (b) Find g(g(5)). .................................................... [2] (c) Solve f(h(x)) = 10. x = ......................... or x = ......................... [3] (d) Find g(h(f(x))) in terms of x. .................................................... [2]

Mark scheme: Question Answer Marks Partial Marks 1(a)(i) 10 1 1(a)(ii) 0.1 1 1(b) 5 2 M1 for g(5) = 0.2 oe 1(c) 0 and –4 nfww 3 M1 for h(x) = 4 or 3( x + 2) 2 − 2[ = 10] B1 for ( x + 2) 2 = 4 oe or 3 x 2 + 12 x = 0 oe 1(d) 1 1 2 M1 for (3 x − 2 + 2) 2 or oe final answer 9x 2 ( 3x ) 2

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2 Alan, Brendan and Cieran work as gardeners. (a) The total amount of money they earn is shared in the ratio of the time each person works. One day Alan works for 2 hours 40 minutes, Brendan works for 5.5 hours and Cieran works for 200 minutes. They earn, in total, $379.50 . By changing all the times into minutes, find the amount of money each person earns. Alan $ ................................................... Brendan $ ................................................... Cieran $ ................................................... [5] (b) (i) Alan needs to buy some gardening tools. In shop A, the price of the tools is $70.20 . In shop B, the price of the tools is 5% less than in shop A. Find the price of the tools in shop B. $ .................................................... [2] (ii) The price of $70.20 is 8% higher than it was last year. Find the price last year. $ .................................................... [3] (c) (i) Brendan invests $450 for 5 years at a rate of 3.5% per year simple interest. Show that the total value of this investment after 5 years is $528.75 . [2] (ii) Cieran invests $450 for 5 years at a rate of x % compound interest. The value of Cieran’s investment after 5 years is $530.60 . Find the value of x. x = .................................................... [3]

Mark scheme: 2(a) 88, 181.5, 110 5 B4 for any two correct or all three correct values seen OR M1 for converting times to same units e.g. 160 : 330 : 200 M2 for correct method to find any part their 160or330or200 e.g. × 379.5 oe their 690 or M1 for correct use of total 379.5 e.g. soi 0.55 their 690 A1 for any one value correct, correctly placed 2(b)(i) 66.69 2 M1 for 70.2 × 0.95 oe 2(b)(ii) 65[.00] cao 3 70.2 M2 for oe 1.08 or M1 for 70.2 = 108% soi 2(c)(i) 3.5 3.5 M1 450 × [ × 5] or 5 × [ × 450] or 100 100 better 3.5 A1 i.e. full and correct conclusion to 450 + 78.75 450 + 450 × 5 × [ = 528.75] 100 leading to 450 + 78.75 or better. 2(c)(ii) 3.35 or 3.350… 3 530.6 M2 for 5 450 or M1 for 450 × [ ]5 = 530.6 oe

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3 Pepe wants to find out if there is a correlation between the hours of sunshine, x hours, and the rainfall, y cm, in Phuket. Pepe recorded the following results. Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec Daily sunshine 8 9.2 7.9 9.4 8 7.4 7.9 8 7.3 7.4 7.5 8 (x hours) Monthly rainfall 4 3 4 15 20 24 30 26 40 28 20 6 (y cm) (a) (i) Complete the scatter diagram. The first eight points have been plotted for you. y 45 40 35 30 25 Monthly rainfall (cm) 20 15 10 5 0 x 7 8 9 10 Daily sunshine (hours) [2] (ii) What type of correlation is shown by the scatter diagram? .................................................... [1] (b) (i) Find the mean number of hours of sunshine. .................................................... hours [1] (ii) Find the mean rainfall. .................................................... cm [1] (c) (i) Find the equation of the regression line for y in terms of x. y = .................................................... [2] (ii) Estimate the rainfall when the number of hours of sunshine is 7.7 . .................................................... cm [1]

Mark scheme: 3(a)(i) Points correctly plotted 2 B1 for 2 or 3 correct points 3(a)(ii) Negative 1 3(b)(i) 8 1 3(b)(ii) 1 1 18.3 or 18.33 or 183 3(c)(i) y = 97 [.0 ] − 9.84 x 2 or 97.02... and –9.836... B1 for 97[.0] + kx, or a – 9.84x, If 0 scored SC1 for 97 – 9.8x 3(c)(ii) 21.2 to 21.3 or 21 1 Strict FT their (c)(i) provided a linear expression

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4 The masses of 120 peaches are recorded in the table. Mass (m grams) Frequency 0 1 m G 120 12 120 1 m G 150 27 150 1 m G 180 33 180 1 m G 210 15 210 1 m G 250 28 250 1 m G 300 5 (a) Calculate an estimate of the mean mass of a peach. Give your answer correct to the nearest gram. .................................................... g [3] (b) Two peaches are chosen at random. Find the probability that they both have a mass of more than 210 g. Give your answer as a fraction in its simplest form. .................................................... [3] (c) (i) Complete the frequency density column in this table. Frequency Mass (m grams) Frequency density 0 1 m G 120 12 120 1 m G 150 27 150 1 m G 180 33 180 1 m G 210 15 210 1 m G 250 28 250 1 m G 300 5 0.1 [2] (ii) On the grid, draw an accurate histogram to show this information. Frequency density 0 m 0 50 100 150 200 250 300 Mass (grams) [4]

Mark scheme: 4(a) 171 cao nfww 3 B2 for 171.25 or 171.3 or M2 for complete method with 1 numerical error or M1 for at least 3 mid-pts (60, 135, 165, 195, 230, 275) soi 4(b) 44 3 1056 cao B2 for oe accept 0.0739 or 0.07394 to 595 14280 0.07395 33 32 or M1 for × 120 119 4(c)(i) 0.1, 0.9, 1.1, 0.5, 0.7, [0.1] 2 B1 for 3 or 4 correct 4(c)(ii) Correct histogram 4 B1 for suitable scale B1 for correct column widths B1FT for 4 or more correct heights

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5 A NOT TO SCALE E D F B C ABC is a triangle and BCFD is a parallelogram. 1 1 AD = AB and A E = A C . 3 3 AB = 6p and AC = 6q . (a) Find an expression, in terms of p and/or q, for (i) BC, .................................................... [1] (ii) DE, .................................................... [2] (iii) FC, .................................................... [1] (iv) BE. .................................................... [2] (b) The area of triangle ADE is 24 units2. (i) Find the area of triangle ABC. .................................................... units2 [2] (ii) Find the area of triangle EFC. .................................................... units2 [3]

Mark scheme: 5(a)(i) –6p + 6q oe 1 5(a)(ii) –2p + 2q oe 2 FT their (a)(i) ÷ 3 provided in form ap +bq B1 for –2p + kq or for kp + 2q JJJG JJJG M1 for AD = 2p oe or AE = 2q or correct route 5(a)(iii) 4p cao 1 5(a)(iv) –6p +2q oe 2 B1 for –6p + kq or for kp + 2q M1 for a correct route 5(b)(i) 216 2 2  1  2 M1 for   or 3 oe soi  3  5(b)(ii) 96 3 2  1  2 M2 for   or 2 oe soi  2  or M1 for triangle EFC is similar to triangle EDA soi

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6 V NOT TO SCALE 8 cm B C √72 cm P A D √72 cm The diagram shows a pyramid with a square base ABCD of side 72 cm. The diagonals of the base, AC and BD, meet at P. The vertex, V, is vertically above P and VP = 8 cm. (a) Find the volume of the pyramid. Give the units of your answer. .................................. ................ [3] (b) Find the length AC. AC = .................................................... cm [2] (c) Find the length DV. DV = .................................................... cm [3] (d) Find angle VDP. Angle VDP = .................................................... [2] (e) X is the midpoint of the side CD. (i) Find the length VX. VX = .................................................... cm [3] (ii) Find angle VXP. Angle VXP = .................................................... [2] (f) The pyramid is cut parallel to ABCD to form a smaller pyramid VEFGH. The volume of VEFGH is 24 cm3. Find the vertical height of this pyramid. .................................................... cm [3]

Mark scheme: 6(a) 192 2 1 2 M1 for × 72 × 8 oe ( ) 3 cm3 1 6(b) 12 2 M1 for ( 72) 2 + ( 72) 2 oe 6(c) 10 3 2 2 M2 for 8 + ( 0.5 their (b) ) or M1 for [PD oe =] 0.5 × their (b) (d) 53.1 or 53.13 2 8 8 M1 for tan = or sin = 0.5 × their (b) their (c) 0.5 × their (b) or cos = their (c) 6(e) (i) 82 or 9.06 or 9.055... 3 M2 for 8 2 + (0.5 × 72) 2 or (their (c))2 – (0.5 × 72) 2 or M1 for (0.5 × 72) 2 6(e)(ii) 62.1 or 62[.0] or 62.00 to 62.10 2 8 M1 for tan = oe 0.5 × 72 6(f) 4 cao 3 24 their (a) M2 for 3 or 3 their (a) 24 1 soi by 2 or 2 24 their (a) or M1 for or their (a) 24 1 soi by 8 or 8

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7 30 cm 10 cm The diagram shows a hollow metal hemisphere. The outside diameter of the hemisphere is 30 cm and the inside diameter is 10 cm. (a) Find the volume of metal used to make the hemisphere. .................................................... cm3 [3] (b) Find the total surface area of the hemisphere. .................................................... cm2 [5]

Mark scheme: 7(a) 6810 or 6806 to 6808 3 1 4 3 3 M2 for × π(15 − 5 ) 2 3 1 4 3 1 4 3 or M1 for either [ ×] π × 15 or [ ×] π × 5 2 3 2 3 7(b) 2200 or 2199… 5 M4 for 2 × π× 5 2 + 2 × π× 15 2 + π× (15 2 − 5 2 ) or M1 for each term

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8 You may use the grid to help you in answering this question. The transformation P is a rotation through 90° anti-clockwise about the origin. The transformation Q is a reflection in the line y =- x . (a) Find the image of the point (5, 1) under the transformation P. (................ , ...............) [2] (b) Find the image of the point (5, 1) under the transformation Q. (................ , ...............) [2] (c) Describe fully the single transformation equivalent to the transformation P followed by the transformation Q. ................................................................................................................................................................... .............................................................................................................................................................. [2]

Mark scheme: 8(a) (–1, 5) 2 B1 for each 8(b) (–1, –5) 2 B1 for each 8(c) Reflection 2 B1 for each y-axis oe

More questions on Transformations

9 y 12 0 x 4 –2 f (x) = 10 + x - x 2 for 0 G x G 4 (a) (i) On the diagram, sketch the graph of y = f (x) . [2] (ii) Write down the co-ordinates of the points where the graph crosses the axes. (................ , ...............) or (................ , ...............) [2] (iii) Solve f (x) = 1. x = .................................................... [1] (b) g (x) = x 2 - 10 log x (i) On the same diagram, sketch the graph of y = g (x) , for 0 1 x G 4 . [2] (ii) Write down the co-ordinates of the minimum point of g (x). (................ , ...............) [2] (iii) Solve the equation. f (x) = g (x) .................................................... [2] (iv) Solve the equation. f (x - 1) = g (x - 1) .................................................... [2]

Mark scheme: 9(a)(i) Correct graph 2 B1 for correct shape with a max 9(a)(ii) (0, 10) 2 B1 for each (3.7[0], 0) or (3.701 to 3.702, 0) 9(a)(iii) 3.54 or 3.541… 1 9(b)(i) Correct graph 2 B1 for correct shape with a min 9(b)(ii) (1.47, 0.488) or (1.473 to 1.474, 2 B1 for each 0.4877…) 9(b)(iii) 0.0982 or 0.09819 to 0.09820 2 B1 for each and 2.98 or 2.975 or 2.976 9(b)(iv) 1.1[0] or 1.098… 2 FT their (iii) + 1 3.98 or 3.975 to 3.976 B1 for each

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10 (a) Solve the equation 4x 2 = 12 - 3x . Give your answers correct to 2 decimal places. You must show all your working. x = ......................... or x = ......................... [4] (b) Solve the inequality 4x 2 2 12 - 3x . .................................................... [2] (c) Solve the inequality 4x 2 + 5 G 12 - 3x . .................................................... [4]

Mark scheme: 10(a) appropriate sketch giving one M2 M1 for sketch of parabola or parabola and positive and one negative answer or 2 −±3 ...... fully correct use of formula straight line or 3 − 4(4)( −12) or 2(4) oe 1.4[0] and –2.15 final answers B2 B1 for each If 0 scored B1 for 1.397… and –2.147... or SC1 for 2.15 and –1.4[0] 10(b) x > 1.40 and x < −2.15 2 FT [ x ] > their max(a), [ x ] < their min(a) B1 for each 10(c) −1.75 - x - 1 nfww 4 B3 for 1, − 1.75 oe B2 for 1 inequality correct B1 for 1 correct value seen or M2 for appropriate sketch or correct factorising or correct use of formula or M1 for 4 x 2 + 3 x − 7 - 0

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11 (a) Solve the simultaneous equations. You must show all your working. 3x - 2y = 11 4x - 5y = 10 x = .................................................... y = .................................................... [4] (b) Use your answers to part (a) to solve the simultaneous equations. 3a - 2b = 22 4a - 5b = 20 a = .................................................... b = .................................................... [2] (c) (i) Use your answers to part (a) to find the exact answers to these simultaneous equations. 3 # 10 p - 2 # 10 q = 11 4 # 10 p - 5 # 10 q = 10 p = .................................................... q = .................................................... [3] (ii) Find the value of p + q . .................................................... [1]

Mark scheme: 11(a) [x = ] 5 4 M1 for correctly equating one set of coefficients [y = ] 2 with correct working M1 for correct method to eliminate one variable OR M1 for equation x = or y = from one equation M1 for correct substitution into other equation B1 for x = 5 B1 for y = 2 If zero scored SC1 for correct subst into one of original equs and evaluation to find other variable 10(b) [ a = ]10 2 B1 for each FT their (a) × 2 [b = ] 4 10(c)(i) [ p = ] log 5 and [ q = ] log 2 3 B2 FT their (a) for either seen Final answers or B1 FT for each correct decimal answer 0.699 or 0.6989 to 0.6990 0.301 or 0.3010... or M1 for 10 p = their 5 or 10 q = their 2 10(c)(ii) 1 cao 1

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