# Use the substitution u x 2 to find x3 dx x 2

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Unformatted text preview: itions for S, find their coordinates. (15) (Total 29 marks) 786. Bag A contains 2 red and 3 green balls. (a) Two balls are chosen at random from the bag without replacement. Find the probability that 2 red balls are chosen. (2) Bag B contains 4 red and n green balls. (b) Two balls are chosen without replacement from this bag. If the probability that two red 2 balls are chosen is , show that n = 6. 15 (4) A standard die with six faces is rolled. If a 1 or 6 is obtained, two balls are chosen from bag A, otherwise two balls are chosen from bag B. (c) Calculate the probability that two red balls are chosen. (4) (d) Given that two red balls are chosen, find the probability that a 1 or a 6 was obtained on the die. (3) (Total 13 marks) 541 787. The continuous random variable X has probability density function 1 x(1 + x2) for 0 ≤ x ≤ 2, 6 f(x) = 0 otherwise. f(x) = (a) Sketch the graph of f for 0 ≤ x ≤ 2. (2) (b) Write down the mode of X. (1) (c) Find the mean of X. (4) (d) Find the median of X. (5) (Total 12 marks) 788. Use mathematical induction to prove that 5n + 9n + 2 is divisible by 4, for n ∈ + . (Total 9 marks) 789. Consider the complex geometric series eiθ (a) 1 2iθ 1 3iθ e + e +… 2 4 Find an expression for z, the common ratio of this series. (2) (b) Show that z< 1. (2) (c) Write down an expression for the sum to infinity of this series. (2) 542 (i) Express your answer to part (c) in terms of sin θ and cos θ. (ii) (d) Hence show that cos θ + 1 1 4 cos θ − 2 cos 2θ + cos 3θ +…= 2 4 5 − 4 cos θ (10) (Total 16 marks) 790. When a fair die is thrown, the probability of obtaining a ‘6’ is 1 . 6 Charles throws such a die repeatedly. (a) Calculate the probability that (i) he throws at least two ‘6’s in his first ten throws; (ii) he throws his first ‘6’ on his fifth throw; (iii) he throws his third ‘6’ on his twelfth throw. (10) (b) On which throw is he most likely to throw his first ‘6’? (2) (Total 12 marks) 791. In an opinion poll, 540 out of 1200 people interviewed stated that they support government policy on taxation. (a) (i) Calculate an unbiased estimate of the proportion, p, of the whole population supporting this policy. (ii) Calculate the standard error of your estimate. (iii) Calculate a 95 % confidence interval for p. (9) (b) State an assumption required to find this interval. (2) (Total 11 marks) 543 792. The 10 children in a class are given two jigsaw puzzles to complete. The time taken by each child to solve the puzzles was recorded as follows. Child A B C D E F G H I J Time to solve Puzzle 1 (mins) 10.2 12.3 9.6 13.8 14.3 11.6 10.5 8.3 9.3 9.9 Time to solve Puzzle 2 (mins) 11.7 12.9 9.9 13.6 16.3 12.2 12.0 8.4 9.8 9.5 (a) For each child, calculate the time taken to solve Puzzle 2 minus the time taken to solve Puzzle 1. (2) (b) The teacher believes that Puzzle 2 takes longer, on average, to solve than Puzzle 1. (i) State hypotheses to test this belief. (ii) Carry out an appropriate t-test at the 1 % significance level and state your conclusion in the context of the problem. (11) (Total 13 marks) 793. Let X1, X2, …, X12 be a random sample from a continuous uniform distribution defined...
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## This note was uploaded on 06/24/2013 for the course HISTORY exam taught by Professor Apple during the Fall '13 term at Berlin Brothersvalley Shs.

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