Cambridge IGCSE Mathematics - International 0607 — 2023 May/June Paper 4 · Variant 2

1 For each of these sequences, find the next term and an expression for the nth term. (a) 17 14 11 8 5 … next term ................................................ nth term ................................................ [3] 1 2 3 4 5 (b) … 2 3 4 5 6 next term ................................................ nth term ................................................ [2] (c) 4 8 16 32 64 … next term ................................................ nth term ................................................ [3] (d) - 2 5 24 61 122 … next term ................................................ nth term ................................................ [3]

Mark scheme: Question Answer Marks Partial Marks 1(a) 2 1 20 – 3n oe final answer 2 M1 for k – 3n or for correct answer seen but then spoiled 1(b) 6 1 7 n 1 oe final answer n  1 1(c) 128 1 2n + 1 oe final answer 2 M1 for 2n + k oe 1(d) 213 1 n3 – 3 oe final answer 2 B1 for any cubic seen or M1 for third differences = 6 or for correct answer seen but then spoiled

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2 The population of a species of bird is estimated to be decreasing by 4% per year. At the end of 2020 the population was 4.32 million. (a) Find the population at the end of 2019. ..................................... million [2] (b) Calculate an estimate for the population at the end of 2025. ..................................... million [2] (c) Find the year in which the population is first expected to be below 2 million. ................................................. [4]

Mark scheme: 2(a) 4.5 nfww 2 4  M1 for P  1    4.32 oe or better  100  2(b) 3.52 or 3.522... nfww 2 5  4  M1 for 4.32 1  oe    100  2(c) 2039 nfww 4 B3 for answer 18.9 or 18.86 to 18.87 or 19 nfww OR  4   2  M3 for n log  1   = log   oe  100   4.32  or good sketch indicating value between 18 and 19 or correct trials as far as 18 and 19  4  n 2 or M2 for 1  = oe    100  4.32 or sketch that could lead to solution or at least 3 correct trials  4  n or M1 for 4.32 × 1  = 2 soi.    100  or at least 2 correct trials

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3 y 9 8 7 6 5 C 4 3 B A 2 1 x 0 – 6 – 5 – 4 – 3 – 2 – 1 1 2 3 4 5 6 – 1 – 2 – 3 – 4 – 5 – 6 – 7 (a) Reflect triangle A in the line y =- 1. [2] - 5 (b) Translate triangle A by the vector [2] e 3o. (c) Describe fully the single transformation that maps triangle A onto triangle B. ..................................................................................................................................................... ..................................................................................................................................................... [3] (d) Describe fully the single transformation that maps triangle A onto triangle C. ..................................................................................................................................................... ..................................................................................................................................................... [3]

Mark scheme: 3(a) Triangle at (1, –6), (2, –4), (4, –4) 2 B1 for reflection in x = – 1 or y = k 3(b) Triangle at (–4, 7), (–3, 5), (– 1, 5) 2   5  k B1 for translation   or   k  3 3(c) Rotation 3 B1 for each 90˚ [anticlockwise] oe [Centre] origin or (0, 0) 3(d) Stretch 3 B1 for each [Factor] 2 x-axis (or y = 0) invariant

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4 The masses, m kg, of 160 students are recorded in the table. Mass, m kg 40 1 m G 50 50 1 m G 60 60 1 m G 70 70 1 m G 80 80 1 m G 90 90 1 m G 100 Frequency 6 18 66 40 18 12 (a) Draw a cumulative frequency curve for these results. 160 140 120 100 Cumulative frequency 80 60 40 20 0 m 40 50 60 70 80 90 100 Mass (kg) [4] (b) Use your cumulative frequency curve to estimate (i) the median ............................................. kg [1] (ii) the interquartile range. ............................................. kg [2] (c) The masses of 60% of the students lie in the range p kg 1 m kg G 80 kg . Use your cumulative frequency curve to estimate the value of p. p = ................................................. [3]

Mark scheme: 4(a) Correct cumulative frequency curve 4 B2 for 5 or 6 correct heights or B1 for 6, 24, 90, 130, 148, 160 soi B1 for points plotted at right hand end of interval. 4(b)(i) 66 to 69 1 FT their curve 4(b)(ii) 10 to 15 2 B1 for [lq] = 62 to 64 or [uq] = 74 to 77 FT their curve for B1 4(c) 61 to 63 nfww 3 B2 for 34 soi or M1 for 160 × 0.6 oe soi by 96

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5 (a) The diagram shows a regular pentagon with sides of 10 cm and centre O. B 10 cm A C NOT TO O SCALE E D (i) Find angle AOB. Angle AOB = ................................................. [1] (ii) Show that OA = 8. 51 cm correct to 3 significant figures. [3] (iii) Find the area of the pentagon. .......................................... cm2 [2] (b) V NOT TO 18 cm SCALE B A C O E 10 cm D The regular pentagon in part (a) is the base of a pyramid. The sloping edges, VA, VB, VC, VD, and VE, are each of length 18 cm. (i) Calculate the perpendicular height, VO, of the pyramid. VO = ............................................ cm [3] (ii) Calculate the volume of the pyramid. .......................................... cm3 [2] (iii) A geometrically similar pyramid has volume 1500 cm 3. Calculate the length of a side of the base of this pyramid. ............................................ cm [3]

Mark scheme: 5(a)(i) 72 1 5(a)(ii) 5 M2  1 5 oe oe M1 for sin  their 72   1   2  OD sin  their 72   2  8.506 to 8.507 A1 5(a)(iii) 172 or 172.0 to 172.2 2 1 M1 for × 8.512 × sin(their 72) oe 2 1 or  8.51  (10or5)sin54 oe 2 1 or  (10or5)  5tan54 oe 2 5(b)(i) 15.9 or 15.86... 3 M2 for 182 – 8.512 or M1 for VO2 + 8.512 = 182 5(b)(ii) 909 to 913 2 1 M1 for  (their172)  (their15.9) 3 5(b)(iii) 11.8 or 11.79 to 11.82 3 1500 M2 for 10 × 3 oe their(b)(ii) 1500 their(b)(ii) or M1 for 3 or 3 their(b)(ii) 1500 their (b)(ii)  10  3 or   oe 1500  x 

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6 y 9 x – 6 0 6 – 9 x 2 + 3 x f ( x) = ( x - 2)( x + 1) (a) On the diagram sketch the graph of y = f ( x) for values of x between - 6 and 6. [3] (b) Write down the equations of the asymptotes parallel to the y‑axis. ............................................................................... [2] (c) Find the zeros of the graph of y = f ( x) . ................................................. [2] (d) g ( x) = x - 3 (i) On the diagram sketch the graph of y = g ( x) for - 6 G x G 6 . [1] (ii) Use your graphs to solve f ( x) = g ( x) . .............................................................................. [3] (iii) Solve g ( x) 2 f ( x) . ............................................................................................................................................. [3]

Mark scheme: 6(a) Correct Sketch 3 B1 for each branch correct 6(b) x = 2, x = –1 2 B1 for each 6(c) –3, 0 2 B1 for each 6(d)(i) Correct Sketch 1 6(d)(ii) –1.16 or –1.162... 3 B1 for each 1 5.16 or 5.162... 6(d)(iii) –1.16 < x < –1 3 B1FT from their (d)(ii) and their (b) for each. 1< x < 2 FT dep on answers to (d)(ii) that lead to three x > 5.16 equivalent inequalities Same accuracy as (d)(ii)

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7 A is the point ( - 8 , 2) and C is the point (8, 10). y C NOT TO SCALE A x O (a) Find the equation of the line AC. ................................................. [3] (b) N is the point (4, 8). Show that N lies on AC. [1] (c) Find the equation of the line that is perpendicular to AC and passes through N. ................................................. [3] (d) A and C are two vertices of a quadrilateral ABCD. B is the point (2, 12). D is the reflection of B in the line AC. (i) Find the coordinates of D. ( ...................... , ...................... ) [2] (ii) Write down the name of the special quadrilateral ABCD. ................................................. [1] (iii) Find the length AC. ................................................. [2] (iv) Find the area of the quadrilateral ABCD. ................................................. [3]

Mark scheme: 7(a) 1 3 1 y = x + 6 oe final answer B2 for x + 6 2 2 OR 10  2 M1 for oe 8 ( 8) M1 for substituting (–8, 2) or (8, 10) into y = (their m)x + c oe 7(b) 1 1 × 4 + 6 = 8 oe 2 7(c) y = –2x + 16 oe final answer 3 B2 for –2x + 16 OR 1 M1 for grad =  1 their 2 M1 for substituting (4, 8) into y = (their m)x + c 1 oe, their m ≠ their 2 7(d)(i) (6, 4) 2 B1 for each coordinate 7(d)(ii) Kite 1 7(d)(iii) 2 M1 for (8 – (–8))2 + (10 – 2)2 17.9 or 17.88 to 17.89 or 8 5 oe 7(d)(iv) 80 or 79.5 to 80.5 3 1 M2 for × their (d)(iii) × their BD 2 1 or 2 × × their (d)(iii) × their BN oe 2 i.e. a correct method for the area of ABCD. or B1for [BN =] 4.47 or 4.472... or 2 5 oe or [BD =] 8.94 or 8.944... or 4 5 oe or M1 for a correct method for the area of one of the triangles in ABCD.

More questions on Transformations

8 A ship sails from port A at a constant speed of 18 km/h on a bearing of 040°. A motorboat sails in a straight line at a constant speed from port B to intercept the ship. Port B is 30 km due south of port A. The ship leaves port A at 08 20 and the motorboat leaves port B at 08 30. The motorboat intercepts the ship at point C at 09 50. North C NOT TO SCALE 40° A 30 km B (a) Find the speed of the motorboat. ......................................... km/h [5] (b) Find the bearing on which the motorboat sails. ................................................. [3]

Mark scheme: 8(a) 40.2 or 40.17 to 40.19 5 B1 for 27 M2 for 30 2  (their 27) 2  2  30  ( their 27)  cos140 oe or M1 for 302 + (their27)2 – 2×30×(their27)×cos 140 M1 for their 53.57 ÷ their time from B to C 8(b) [0]18.9 or [0]18.89 to [0]18.91 3 their 27sin140 M2 for oe their 53.57 sin sin140 or M1 for  their 27 their 53.57

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9 P E R C E N T I L E Asa and Bernice have these 10 letter cards. A, E, I, O and U are vowels. All other letters are consonants. (a) Asa picks a card at random. Write down the probability that Asa’s card shows the letter T. ................................................. [1] (b) Asa replaces his card. Bernice picks two cards at random without replacement. Calculate the probability that both of Bernice’s cards are vowels. ................................................. [2] (c) Bernice replaces her cards. Asa picks 3 cards at random without replacement. Calculate the probability that Asa’s cards can be arranged to spell the word PEN. ................................................. [3] (d) Asa replaces his cards. Bernice picks cards at random with replacement until she first gets a consonant. 48 The probability that she first gets a consonant on her nth pick is . 3125 Find the value of n. ................................................. [3]

Mark scheme: 9(a) 1 1 oe 10 9(b) 2 2 4 3 oe M1 for  15 10 9 9(c) 1 3 1 3 1 oe M2 for k ×   oe, k = 3, 4, 5 or 6 40 10 9 8 1 3 1 or M1 for   oe 10 9 8 If 0 scored SC1 for indicating 6 possibilities 9(d) 5 3 n1  4   6  48 M2 for       oe  10   10  3125  4  k  6  or M1 for      , k ⩾ 2 oe  10   10 

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10 (a) Simplify. 3x - 5y + 4x - 6y ................................................. [2] (b) Expand. x ( x + 2) ................................................. [1] (c) Factorise. 10ab + 8ac - 15b 2 - 12bc ................................................. [2] 2 5 (d) - = 3 2x + 1 x - 3 (i) Show that 6x 2 - 7x + 2 = 0 . [4] (ii) Solve 6x 2 - 7x + 2 = 0 . You must show all your working. x = .................. or x = .................. [3]

Mark scheme: 10(a) 7x – 11y Final answer 2 B1 for 7x – ky or kx – 11y k not zero 10(b) x2 + 2x Final answer 1 10(c) (5b + 4c)(2a – 3b) Final answer 2 M1 for 2a(5b + 4c) – 3b(5b + 4c) or 5b(2a – 3b) + 4c(2a – 3b) or correct answer seen but spoiled 10(d)(i) 2(x – 3) – 5(2x +1) = 3(2x + 1)(x – 3) M1 oe or better 2x – 6 – 10 x – 5 or better B1 [3](2x2 – 6x + x – 3) oe or better B1 completion to 6x2 – 7x + 2 [ = 0] A1 at least one step with no errors or omissions 10(d)(ii) (2x – 1)(3x – 2) [= 0] M2 M1 for pair of brackets giving two terms or sketch of parabola showing two correct positive solutions or sketch of any parabola for +ve x2 2 ( 7) ( 7)  (  7)  4(6)(2) or correct formula with or or 2  6 2  6 ( 7) 2  4(6)(2) seen 1 2 B1 , oe 2 3

More questions on Algebraic fractions

11 f ( x) = 2 x + 5 g ( x) = 1 - 3 x (a) Find f ( - 2) . ................................................. [1] (b) Solve f ( g ( x)) = 19 . ................................................. [3] (c) Find g -1 ( x) . g -1 ( x) = ................................................. [2] g ( x) (d) y = f ( x) Find x in terms of y. x = ................................................. [3]

Mark scheme: 11(a) 1 1 11(b) –2 3 B2 for –6x = 12 oe or better or M1 for 2(1 – 3x) + 5 = 19 11(c) 1  x 2 y 1 oe Final answer M1 for x = 1 – 3y or y + 3x = 1 or   x 3 3 3 or y – 1 = –3x 11(d) 1  5 y 3 M1 for y(2x + 5) = 1 – 3x oe oe Final answer M1FT dep for 2xy + 3x = 1 – 5y dependent on 2 y  3 4 term equation with 2 terms in x. M1FT for factorising and dividing to form a  by c  dy Max 2 marks if final answer is incorrect.

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