Cambridge A Level Mathematics 9709 — 2023 May/June Paper 5 · Variant 2

2 A sports event is taking place for 4 days, beginning on Sunday. The probability that it will rain on Sunday is 0.4. On any subsequent day, the probability that it will rain is 0.7 if it rained on the previous day and 0.2 if it did not rain on the previous day. (a) Find the probability that it does not rain on any of the 4 days of the event. 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(b) Find the probability that the first day on which it rains during the event is Tuesday. 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(c) Find the probability that it rains on exactly one of the 4 days of the event. 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Mark scheme: 2(a) [P(no rain) =   3 0.6 0.8  =] 0.3072, 192 625 1 Question Answer Marks Guidance 2(b) 0.6 0.8 0.2   M1 a  b  c where a, b = 0.6, 0.8, c = 0.2, 0.4, 0.7 . Condone including Wednesday with both 0.3 and 0.7 used. = 0.096[0], 12 125 A1 2 2(c)   P RDDD 0.4 0.3 0.8 0.8     = 0.0768, 48 625   P DRDD 0.6 0.2 0.3 0.8     = 0.0288, 18 625   P DDRD 0.6 0.8 0.2 0.3     = 0.0288, 18 625   P DDDR 0.6 0.8 0.8 0.2     = 0.0768, 48 625 B1 Correct probability for one clearly identified outcome evaluated accept unsimplified. A correct unsimplified expression is not sufficient. M1 Add 4 probability values, 0 < p < 1, for appropriate identified scenarios. Accept unsimplified. Ways of identifying scenarios for this mark: Stating the days. All the unsimplified probability calculations exactly as stated in the mark scheme. Identifying the correct branches on a tree diagram and linking with the values. No repeated scenarios. No incorrect scenarios. 0.2112, 132 625 A1 Accept 0.211 If 0/3 scored SC B1 for 0.2112, 132 625. 3

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3 The following back-to-back stem-and-leaf diagram represents the monthly salaries, in dollars, of 27 employees at each of two companies, A and B. Company A Company B 5 4 1 1 0 25 4 4 5 6 6 7 9 9 8 7 2 1 0 26 0 1 3 5 5 7 9 9 8 6 4 2 1 0 27 1 3 4 6 6 8 8 6 5 4 2 0 28 0 1 2 2 2 9 8 5 29 1 30 9 A and $2760 for company B Key: 1 . 27 . 6 means $2710 for company (a) Find the median and the interquartile range of the monthly salaries of employees in company A. [3] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ The lower quartile, median and upper quartile for company B are $2600, $2690 and $2780 respectively. (b) Draw two box-and-whisker plots in a single diagram to represent the information for the salaries of employees at companies A and B. [3] (c) Comment on whether the mean would be a more appropriate measure than the median for comparing the given information for the two companies. [1] ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................ ........................................................................................................................................................

Mark scheme: 3(a) Median = 2710 B1 Must be identified, condone Q2. Ignore units throughout. 2840 – 2610 M1 2820 ⩽ UQ ⩽ 2850 – 2600 ⩽ LQ ⩽ 2620. 230 A1 www If M0 scored SC B1 for 230 www. If key ignored consistently: B0 Median = 271 SC M1 282 ⩽ UQ ⩽ 285 – 260 ⩽ LQ ⩽ 262 SC A1 23. 3 3(b) Box-and-whisker plot on provided grid. B1 All 5 key values for B plotted accurately in standard format using a linear scale with 3 identified values. Labelled B. Scale at least 1 cm = $100. B: 2540 2600 2690 2780 3090 A: 2500 2610 2710 2840 3010 B1FT All 5 key values for A, FT from (a), plotted accurately in standard format using a linear scale with 3 identified values. Labelled A. Scale at least 1cm = $100 B1 Whiskers not through box for both, not drawn at corners of boxes, single linear scale for the diagram and labelled ‘salaries’ (oe) and $. 3 Question Answer Marks Guidance 3(c) Examples: Mean less appropriate than median because of extreme value for company B [at $3090]. No, extreme value in company B. No, $3090 is an anomaly. B1 Must refer to company B, may be implied by appropriate use of $3090. Must include an indication that the mean is not appropriate. No contradictory statements can be present, e.g. acceptable comment with ‘but mean could be used for company A’. Condone reference to $309. 1

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5 The lengths of Western bluebirds are normally distributed with mean 16.5cm and standard deviation 0.6cm. A random sample of 150 of these birds is selected. (a) How many of these 150 birds would you expect to have length between 15.4cm and 16.8cm? 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The lengths of Eastern bluebirds are normally distributed with mean 18.4cm and standard deviation 3 cm. It is known that 72% of Eastern bluebirds have length greater than 17.1cm. (b) Find the value of 3. 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A random sample of 120 Eastern bluebirds is chosen. (c) Use an approximation to find the probability that fewer than 80 of these 120 bluebirds have length greater than 17.1cm. 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Mark scheme: 5(a) [P(15.4 < X < 16.8 ) =] P(15.4 16.5 16.8 16.5) 0.6 0.6   Z [= P( 1.833 0.5)    Z ] M1 Use of ± standardisation formula once with 16.5, 0.6 and either 15.4 or 16.8 substituted. [=     Φ 0.5 Φ 1.833 1  ] 0.6915 0.9666 – 1  M1 Calculating the appropriate probability area (leading to their final answer, expect > 0.5). 0.6915 – (1 – 0.9666) or (0.6915 – 0.5) + (0.9666 – 0.5) OE are alternatives. = 0.658 A1 0.658 ≤ p < 0.6585 . If A0 scored, SC B1 for 0.658 ⩽ p < 0.6585 . [Expected number =] 0.6581  150 = 98, 99 B1 FT FT their 4SF (or better) probability from a normal calculation. Must be a positive single integer answer. No approximation notation. 4 5(b) 17.1 18.4 0.72                 P Z 17.1 18.4 0.583    B1 0.5825 < z ⩽ 0.583 or – 0.583 ⩽ z < – 0.5825 seen. M1 Use of the ± standardisation formula with 17.1, 18.4, σ and a z-value (not 0.28, 0.72, 0.4175, 0.2358, 0.7642, 0.6103, 0.3897, …). Condone continuity correct ± 0.05, not 2,  . 2.23  A1 AWRT 3 Question Answer Marks Guidance 5(c) [Mean = 120  0.72 =] 86.4 [Var = 120  0.72  0.28 =] 24.192 B1 86.4, 84 2 5 and 24 24 125 , 24.192 to at least 3SF seen, allow unsimplified. May be seen in standardisation formula. (4.918 ⩽ σ ⩽ 4.919 implies correct variance) Incorrect notation is penalised. P(X < 80) = P( 79.5 86.4) 24.192   Z M1 Substituting their mean (not 18.4) and their positive 4.9185 into ± standardisation formula (any number for 79.5), condone their 4.9182 and √their 4.918 . M1 Using continuity correction 79.5 or 80.5 in their standardisation formula. [P( 1.  Z 4029) = 1−   Φ 1.403 ] 1 − 0.9196 M1 Appropriate area Φ, from final process, must be a probability. Expect final answer < 0.5 . Note: correct final answer implies this M1. 0.0804 A1 0.0803 ⩽ p ⩽ 0.0804 5

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6 In a group of 25 people there are 6 swimmers, 8 cyclists and 11 runners. Each person competes in only one of these sports. A team of 7 people is selected from these 25 people to take part in a competition. (a) Find the number of different ways in which the team of 7 can be selected if it consists of exactly 1 swimmer, at least 4 cyclists and at most 2 runners. 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For another competition, a team of 9 people consists of 2 swimmers, 3 cyclists and 4 runners. The team members stand in a line for a photograph. (b) How many different arrangements are there of the 9 people if the swimmers stand together, the cyclists stand together and the runners stand together? 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(c) How many different arrangements are there of the 9 people if none of the cyclists stand next to each other? 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Mark scheme: 6(a) S + 5C + 1R 6C1  8C5  11C1 [= 6  56  11] 3696  S + 6C [+ 0R] 6C1  8C6 [ 11C0][= 6  28] = 168 B1 Correct outcome/value for 1 identified scenario, accept unsimplified, www. M1 Add values of 3 correct scenarios. No incorrect scenarios, no repeated scenarios. Condone 6Ce  8Cf  11Cg, with e + f + g = 7 to identify S, C, R. [Total =] 26964 A1 cao 4 6(b) 2!  3!  4!  6 M1 2!  3!  4!  k, k an integer > 0. 1 can be implied. =1728 A1 If A0 scored SC B1 for 1728 www. 2 Question Answer Marks Guidance 6(c) Method 1 6!  7  6  5 M1 6!  k, k an integer > 0. 1 can be implied. M1 ! ! ! m a b   7  n  r; 6 ⩽ m ⩽ 9; a = 1, 2; b = 1, 4; 1 ⩽ n, r ⩽ 6, n ≠ r. M1 ! ! ! m a b   7  6  5; 6 ⩽ m ⩽ 9; a =1, 2; b = 1, 4. 151 200 A1 Condone 151 000. If A0 scored SC B1 for 151 200 www. Method 2 6!  7P3 M1 6!  k, k an integer > 0. 1 can be implied. M1 ! ! ! m a b  × 7Pq, or ! ! ! m a b  × 7C !  q q ; 6 ≤ m ≤ 9; a =1, 2; b = 1, 4; 1 ≤ q ≤ 6. M1 ! ! ! m a b  × 7P3, or ! ! ! m a b  × 7 3 C 3!  ; 6 ≤ m ≤ 9; a =1, 2; b = 1, 4. 151 200 A1 Condone 151 000. If A0 scored SC B1 for 151 200 www. Question Answer Marks Guidance 6(c) Method 3 6!  35  3! M1 6!  k, k an integer > 0. 1 can be implied. M1 ! ! ! m a b   35  q!; 6 ⩽ m ⩽ 9; a =1, 2; b = 1, 4; 1 ⩽ q ⩽ 3. M1 ! ! ! m a b   35  6; 6 ⩽ m ⩽ 9; a =1, 2; b = 1, 4. 151 200 A1 Condone 151 000. If A0 scored SC B1 for 151 200 www. Method 4 9! – 7!3! – 3P2  6!  7  6 Or 9! – 7!3! – 3!  7!  6 [= 362 880 – 30 240 – 181 440] M1 9! – 7!r! – q, r an integer > 1, q an integer ⩽ 0. 0 and 1 may be implied. M1 ! ! ! ! s a b c   – 7!3! – q; s = 8, 9; a =1, 2; b = 1, 3; c = 1, 4; q an integer ⩾ 0. 0 and 1 may be implied. M1 ! ! ! ! s a b c   – 7!3! – 3P2  6!  6  7, 6 ⩽ s ⩽ 9, or ! ! ! ! s a b c   – 7!3! – 3!  7!  6, 6 ⩽ s ⩽ 9. a =1, 2 b = 1, 3 c = 1, 4. 1 may be implied. 151 200 A1 Condone 151 000. If A0 scored SC B1 for 151 200 www. 4

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