Cambridge IGCSE Mathematics - International 0607 — 2022 Oct/Nov Paper 4 · Variant 3

1 y 12 11 10 9 8 P 7 6 5 Q T 4 3 2 1 0 1 2 3 4 5 6 7 8 9 10 11 12 x (a) Rotate triangle T through 90° clockwise about the point (9, 6). [2] (b) Enlarge triangle T with scale factor 1, centre (0, 0). [2] 2 (c) Describe fully the single transformation that maps triangle T onto triangle P. ..................................................................................................................................................... ..................................................................................................................................................... [2] (d) Describe fully the single transformation that maps triangle T onto triangle Q. ..................................................................................................................................................... ..................................................................................................................................................... [3]

Mark scheme: Question Answer Marks Partial Marks 1(a) Triangle at (7, 7), (9, 7), (7, 11) 2 B1 for correct orientation but incorrect position or for 90° anticlockwise rotation 1(b) Triangle at (2, 2), (4, 2), (4, 3) 2 B1 for correct size and orientation but incorrect position 1(c) Translation 2 B1 for each  −3     3  1(d) Stretch 3 B1 for each 1 2 x = 1 invariant

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2 y 1 – 5 0 5 x – 3 1 1 f ( x) = - 2 x x (a) On the diagram, sketch the graph of y = f ( x) for values of x between - 5 and 5. [2] (b) Find f ( - 2) . ................................................. [1] (c) Solve the equation f ( x) = 0 . x = ................................................ [1] (d) Find the maximum value of f(x). ................................................. [1] (e) Write down the equation of each asymptote. ..................................................................... [2] (f) (i) Solve the equation. 1 1 2 - = x - 2 2 x x ..................................................................... [3] 1 1 2 4 2 (ii) The equation - 2 = x - 2 can be rearranged to the form x + ax + bx + c = 0 . x x Find the values of a, b and c. a = ................................................ b = ................................................ c = ................................................ [2]

Mark scheme: 2(a) Correct sketch 2 No intersections with y-axis B1 for each branch with no large curl back or feathering. Right hand branch with a maximum or level. 2(b) –[0].75 oe 1 2(c) 1 1 2(d) 0.25 oe 1 Not coordinates 2(e) x = 0 2 B1 for each y = 0 2(f)(i) 0.525 or 0.5248 to 0.5249 3 B2 for one correct or M1 for sketch of y = x2 – 2 added 1.49 or 1.490... to diagram 2(f)(ii) [a =] –2 2 B1 for x −=1 x 4 − 2 x 2 oe [b =] –1 [c =] 1

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3 (a) Amira buys a magazine that costs $n and a book that costs $(2n + 5). She pays with a $20 note and receives $1.62 change. Find the cost of a magazine. $ ................................................ [3] (b) The cost of a bar of chocolate is $x and the cost of a bag of sweets is $y. Bruce buys 2 bars of chocolate and 1 bag of sweets for a total of $3.60 . Charlie buys 3 bars of chocolate and 2 bags of sweets for a total of $6.05 . Find the total cost of 1 bar of chocolate and 3 bags of sweets. You must show all your working. $ ................................................ [5]

Mark scheme: 3(a) 4.46 3 M1 for n + 2n + 5 = 20 – 1.62 oe M1 for an = b from their equation including n and 2n + 5 3(b) 2x + y = 3.6[0] oe B1 3x + 2y = 6.05 oe B1 Correctly eliminating one variable M1 With working seen 5.05 A2 A1 [x =] 1.15 or [y =]1.3[0] If M0 scored, SC1 for their values satisfying one of their original equations

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4 Complete the table for the 5th term and the nth term of each sequence. Sequence 1st term 2nd term 3rd term 4th term 5th term nth term A 3 5 7 9 B 1 8 27 64 C 1 1 1 2 4 2 D 0 2 6 12 [11]

Mark scheme: 4 Sequence A: 11 1 2n + 1 oe final answer 2 B1 for 2n + k or for kn + 1, k ≠ 0 Sequence B: 125 1 n3 oe final answer 1 Sequence C: 4 1 2n – 3 oe final answer 2 B1 for 2n + k oe Sequence D: 20 1 n2 – n oe final answer 2 M1 for 2nd differences = 2 or for any quadratic as final answer

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5 (a) Kris and Laila share $200 in the ratio 2 : 3. (i) Show that Kris receives $80. [1] (ii) Kris spends 30.8% of his $80 on a book. Calculate the cost of the book. $ ................................................ [2] (iii) Laila invests her $120 at a rate of 1.16% per year simple interest. Calculate the total amount Laila has at the end of 5 years. $ ................................................ [3] (b) On 1 January 2020, Sangita invests an amount of money at a rate of 2% per year compound interest. On 1 January 2023 the value of the investment is $5306.04 . (i) Calculate the amount Sangita invested on 1 January 2020. $ ................................................ [2] (ii) Calculate the value of the investment on 1 January 2025. $ ................................................ [2] (c) Tomas invests an amount of money at a rate of 1.4% per year compound interest. Find the number of complete years it takes for the value of his investment to increase by 50%. ................................................. [4]

Mark scheme: 5(a)(i) 2 1 200 2 + 3 5(a)(ii) 24.64 cao 2 30.8 M1 for  80 oe 100 5(a)(iii) 126.96 cao 3 120  1.16  5 M2 for 120 + oe 100 120  1.16  5 or M1 for 100 5(b)(i) 5000 cao 2 3  2  M1 for X  1 + = 5306.04 oe    100  5(b)(ii) 5520.4[0] nfww 2 2  2  M1 for 5306.04  1 + oe    100   2  5 or for their 5000  1 +    100  5(c) 30 nfww 4 B3 for 29.9 or 29.16… OR  1.4   150  or M3 for n log  1 +  = log    100   100  oe or good sketch indicating value between 29 and 30 or correct trials reaching 20 and 21  1.4  n 150 or M2 for 1 + = oe    100  100 or suitable graph with n > 1 or at least 3 correct trials or M1 for  1.4  150    1 +  n =   oe soi  100  100 by at least 2 trials with n > 1

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6 (a) p = r = 4 7 (i) Find 2p. [1] f p 1 (ii) Find p - r . 4 [2] f p (iii) Find the magnitude of p. ................................................. [2] (b) K is the point (3, 4). - 1 (i) The vector from K to L is e 1o. Find the coordinates of L. ( ...................... , ...................... ) [1] 5 (ii) The vector from J to K is e- 2o. Find the coordinates of J. ( ...................... , ...................... ) [1] (c) A is the point ( - 1, 3 ) and B is the point (5, 7). The perpendicular bisector of the line AB meets the x-axis at C. Find the coordinates of C. ( .................... , .................... ) [7]

Mark scheme: 6(a)(i) 4 1  cao 8 6(a)(ii)  1.5  2  1  k   oe cao    B1 for answers oe or 12     −6    6    −   k   1    or for 2 seen      1  6(a)(iii) 2 M1 for 22 + 42 2 5 or 4.47 or 4.472... final answer 6(b)(i) (2, 5) cao 1 6(b)(ii) (–2, 6) cao 1 6(c)  16  7 3  ,0  oe B5 for y = − x + 8 oe  3  2 3 M1 for − x + 8 = 0 oe 2 OR B1 for (2, 5) 7 − 3 M1 for oe (= m1) 5 −−1 1 M1 for grad ( m2 ) = − their m1 M1 for substituting their (2, 5) into y = (their m2) x + c M1 for substituting y = 0 into their equation of line

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7 (a) The time, t hours, spent watching television in one week by each of 100 students is shown in the table. Time, t hours 0 1 t G 10 10 1 t G 20 20 1 t G 25 25 1 t G 30 30 1 t G 60 Frequency 3 11 42 40 4 (i) A pie chart is drawn to show the results. Calculate the sector angle for the number of students who spend more than 30 hours watching television. ................................................. [2] (ii) Calculate an estimate of the mean. .............................................. h [2] (b) A shopkeeper records the midday temperature, t °C, and the number of ice creams, n, sold each day in one week. The table shows the results. Midday 20 24 20 17 18 20 25 temperature, t °C Number of ice 103 106 95 91 93 98 114 creams, n (i) Write down the type of correlation shown in the table. ................................................. [1] (ii) Find the equation of the regression line, giving n in terms of t. n = ................................................ [2] (iii) Use your answer to part(b)(ii) to find the number of ice creams expected to be sold when the midday temperature is 22 °C. ................................................. [1] (iv) During this week, the shopkeeper sells 700 ice creams. She estimates that she will sell a total of 9800 ice creams during the next 14 weeks. Give a reason why this may not be a good estimate. ............................................................................................................................................. [1] (c) When the weather is fine, the probability that Lance goes cycling is 7. 9 When the weather is not fine, the probability that Lance goes cycling is 1. 5 The probability that the weather is fine is 3. 4 (i) Complete the tree diagram. Weather Cycles Yes 7 9 Fine 3 4 No .......... Yes .......... .......... Not fine .......... No [2] (ii) Find the probability that Lance goes cycling. ................................................. [3]

Mark scheme: 7(a)(i) 14.4 2 4 M1 for  360 100 7(a)(ii) 24.05 or 24.1 2 M1 for at least 3 mid-values soi 7(b)(i) positive 1 7(b)(ii) n = 2.61t + 46.3 2 B1 for 2.61t + k or kt + 46.3 or 2.6t + 46 7(b)(iii) 103 or 104 1 FT their (c)(ii) but must be integer answer 7(b)(iv) small sample oe or reference to weather 1 7(c)(i) 1 2 1 4 2 B1 for 2 correct , , , 4 9 5 5 7(c)(ii) 19 3 3 7 1 1 oe M2 FT for  + their  their 30 4 9 4 5 or M1 FT for one of the products only FT probabilities < 1

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8 A E 8 cm NOT TO 16 cm 60° SCALE B 17 cm 32° 18 cm 55° D C The diagram shows a pentagon ABCDE and diagonals BD and BE. (a) (i) Calculate angle BCD. Angle BCD = ................................................ [1] (ii) Calculate BC. BC = ........................................... cm [3] (b) Calculate angle EBD. Angle EBD = ................................................ [3] (c) Calculate the area of the pentagon ABCDE. ......................................... cm2 [4] (d) Calculate the shortest distance from C to AE. ............................................ cm [4]

Mark scheme: 8(a)(i) 93 1 8(a)(ii) 14.8 or 14.76… 3 18sin55 M2 for sin ( their ( i ) ) sin ( their ( i ) ) sin55 or M1 for = oe 18 BC 8(b) 59.7 or 59.65… 3 16 2 + 182 − 17 2 M2 for 2  16  18 or M1 for 172 = 162 + 182 – 2  16  18cos(…) 8(c) 250 or 249.7 to 250.3… 4 1 M1 for 8 16  sin60 oe 2 1 M1 for  16  18  sin theirEBD oe 2 1 M1 for  18  theirBC  sin32 oe 2 8(d) 21[.0] or 21.1 or 20.95 to 21.07 4 Triangle BXC where X is on AB extended and angle BXC = 90° M3 for 8 + their BCcos(180 – 60 – 32 – their(a)) or M2 for their BCcos(180 – 60 – 32 – their(a)) or M1 for angle XBC = 180 – 60 – 32 – their(a) If 0 scored, SC1 for recognition of correct shortest distance, e.g. AX OR M1 for [AC2 =] 82 + (theirBC)2 – 2  8  (theirBC) cos(60 + theirEBD + 32) oe M1 for theirAC theirBC = sin ( 60 + theirEBD + 32 ) sin BAC oe M1 for perp = sin ( 90 − theirBAC ) oe theirAC If 0 scored, SC1 for recognition of correct shortest distance

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9 (a) f ( x) = 2 x + 3 g ( x) = x 2 + 1 h ( x) = 2 sin ( 2x) (i) Find f ( - 2) . ................................................. [1] (ii) Find f -1 ( x) . f -1 ( x) = .............................................. [2] (iii) Find x when g ( x) = 2 f ( x) . x = ............... or x = ............... [3] (iv) Find g ( f ( x)) , giving your answer in the form ax 2 + bx + c . ................................................. [3] (v) Find the amplitude and period of h ( x) . Amplitude = ............................ Period = ............................ [2] (vi) Solve the equation h ( x) = 3 for 0° G x G 180 ° . ................................................. [2] (b) j ( x) = log a x, x 2 0 (i) Find the value of j 3 a ` j. ................................................. [1] (ii) Find j -1 ( x) . j -1 ( x) = ................................................ [2]

Mark scheme: 9(a)(i) –1 1 9(a)(ii) x − 3 2 y 3 oe final answer M1 for y – 3 = 2x or = x + 2 2 2 or x = 2y + 3 9(a)(iii) –1, 5 3 B2 for (x – 5)(x + 1) or sketch indicating –1 and 5 −−( 4 )  ( −4 ) 2 − 4 (1)( −5 ) or oe 2 (1) or M1 for x 2 + 1 = 2(2 x + 3) oe 9(a)(iv) 4x2 + 12x + 10 cao 3 M1 for (2x + 3)2 + 1 B1 for  ( 2 x + 3 ) 2 =  4 x 2 + 6 x + 6 x + 9   or 4 x 2 + 12 x + 9 9(a)(v) 2 2 B1 for each 180 9(a)(vi) 30, 60 2 B1 for each 9(b)(i) 1 1 oe 3 9(b)(ii) ax final answer 2 M1 for a y = x or x = loga y

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10 (a) A machine lays a pipe of length 2.5 km in 18 hours. The machine always works at the same rate. Calculate the time it takes to lay a pipe of length 4 km. ........................................ hours [2] (b) t varies inversely as the square root of x. x varies directly as the square of y. When x = 4, t = 3 . When y = 4, x = 81. ty = h Find the value of h. h = ................................................ [5]

Mark scheme: 10(a) 28.8 2 4 M1 for  18 oe 2.5 10(b) 8 5 6 16 x oe B4 for ty =  oe 3 x 81 2 2 36 16 x or t y =  oe x 81 6 81 2 B3 for t = oe and x = y oe x 16 6 81 2 or B2 for t = oe or x = y x 16 oe k or M1 for t = oe or x = ky2 oe x

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