Cambridge A Level Mathematics 9709 — 2020 May/June Paper 5 · Variant 2

2 A total of 500 students were asked which one of four colleges they attended and whether they preferred soccer or hockey. The numbers of students in each category are shown in the following table. Soccer Hockey Total Amos 54 32 86 Benn 84 72 156 Canton 22 56 78 Devar 120 60 180 Total 280 220 500 (a) Find the probability that a randomly chosen student is at Canton college and prefers hockey. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the probability that a randomly chosen student is at Devar college given that he prefers soccer. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) One of the students is chosen at random. Determine whether the events ‘the student prefers hockey’ and ‘the student is at Amos college or Benn college’ are independent, justifying your answer. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 2(a) 56 14 or or 0.112 500 125 B1 1 2(b) ( ) ( ) ( ) P D S 120 P D|S P S 280 ∩ = = M1 120 3 or 280 7 A1 2 Question Answer Marks 2(c) P(hockey) = 220 0.44 500 = P(Amos or Benn) = 242 0.484 500 = P(hockey ∩ A or B) = 104 0.208 500 = P(H) × P(A U B) = P(H ∩ (A U B)) if independent M1 220 242 1331 500 500 6250 × = so not independent A1 2

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3 Two machines, A and B, produce metal rods of a certain type. The lengths, in metres, of 19 rods produced by machine A and 19 rods produced by machine B are shown in the following back-to-back stem-and-leaf diagram. A B 21 1 2 4 7 6 3 0 22 2 4 5 5 6 8 7 4 3 1 1 23 0 2 6 8 9 9 5 5 5 3 2 24 3 3 4 6 4 3 1 0 25 6 Key: 7 22 4 means 0.227 m for machine A and 0.224 m for machine B. (a) Find the median and the interquartile range for machine A. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ It is given that for machine B the median is 0.232 m, the lower quartile is 0.224 m and the upper quartile is 0.243 m. (b) Draw box-and-whisker plots for A and B. [3] (c) Hence make two comparisons between the lengths of the rods produced by machine A and those produced by machine B. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(a) B1 UQ = 0.245, LQ = 0.231, So IQR = 0.245 – 0.231 M1 0.014 A1 3 Question Answer Marks 3(b) LQ M UQ A 0.220 0.231 FT 0.238 FT 0.245 FT 0.254 B 0.211 0.224 0.232 0.243 0.256 Medians and quartiles correctly plotted for A or B B1 End points correct for A or B B1 Completely correct, including scale B1 3 3(c) Lengths of rods produced by machine A are longer. (B1 for comparison of central tendency) B1 Lengths of rods produced by machine A are less spread out (B1 for comparison of spread) B1 2

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4 Trees in the Redian forest are classified as tall, medium or short, according to their height. The heights can be modelled by a normal distribution with mean 40 m and standard deviation 12 m. Trees with a height of less than 25 m are classified as short. (a) Find the probability that a randomly chosen tree is classified as short. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ Of the trees that are classified as tall or medium, one third are tall and two thirds are medium. (b) Show that the probability that a randomly chosen tree is classified as tall is 0.298, correct to 3 decimal places. 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(c) Find the height above which trees are classified as tall. 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Mark scheme: 4(a) ( ) ( ) ( ) ( ) 25 40 P 25 P P 1.25 P 25 P 12 X z z X z −   < = < = < − < = <     M1 1 – 0.8944 M1 0.106 A1 3 4(b) 0.8944 divided by 3 (M1 for 1 - their (a) divided by 3) M1 0.298 AG A1 2 4(c) 0.2981 gives z = 0.53 B1 40 0.53 12 h − = M1 h = 46.4 A1 3

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6 (a) Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that there is an E at the beginning and an E at the end. [2] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Find the number of different ways in which the 10 letters of the word SUMMERTIME can be arranged so that the Es are not together. [4] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Four letters are selected from the 10 letters of the word SUMMERTIME. Find the number of different selections if the four letters include at least one M and exactly one E. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 6(a) 8! 3! M1 6720 A1 2 Question Answer Marks 6(b) Total number = ( ) ( ) 10! 302400 A 2!3! B1 With Es together = ( ) ( ) 9! 60480 B 3! B1 Es not together = their (A) – their (B) M1 241920 A1 Alternative method for question 6(b) _ ^ _ ^ _ ^ _ ^ _ ^ _ ^ _ ^ _ ^ _ 8! 9 8 3! 2 × × 8! × k in numerator, k integer ≥ 1, denominator ≥ 1 B1 3! × m in denominator, m integer ≥ 1 B1 Their 8! 3! Multiplied by 9C2 (OE) only (no additional terms) M1 241920 A1 4 Question Answer Marks 6(c) Scenarios: E M M M 5C0 = 1 E M M _ 5C1 = 5 E M _ _ 5C2 = 10 M1 Summing the number of ways for 2 or 3 correct scenarios M1 Total = 16 A1 3

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7 On any given day, the probability that Moena messages her friend Pasha is 0.72. (a) Find the probability that for a random sample of 12 days Moena messages Pasha on no more than 9 days. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (b) Moena messages Pasha on 1 January. Find the probability that the next day on which she messages Pasha is 5 January. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ (c) Use an approximation to find the probability that in any period of 100 days Moena messages Pasha on fewer than 64 days. [5] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 7(a) = 1 – [12C100.72100.282 + 12C11 0.72110.281+0.7212] 1 – (0.19372 + 0.09057 + 0.01941) A1 0.696 A1 3 7(b) 0.283 × 0.72 = 0.0158 B1 1 Question Answer Marks 7(c) Mean = 100 × 0.72 = 72 Var = 100 × 0.72 × 0.28 = 20.16 M1 P(less than 64) = P 63.5 72 20.16 z −   <     (M1 for substituting their µ and σ into ±standardisation formula with a numerical value for ‘63.5’) M1 Using either 63.5 or 64.5 within a ±standardisation formula M1 Appropriate area Φ, from standardisation formula P(z<…) in final solution = P(z < –1.893) M1 0.0292 A1 5

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