Y arccos 1 2x2 with respect to x and simplify your

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Unformatted text preview: with four faces is used in a game. A player pays 10 counters to roll the die. The table below shows the possible scores on the die, the probability of each score and the number of counters the player receives in return for each score. Score 1 2 3 4 Probability 1 2 1 5 1 5 1 10 Number of counters player receives 4 5 15 n Find the value of n in order for the player to get an expected return of 9 counters per roll. Working: Answers: ………………………………………….. (Total 4 marks) 36 41. A factory has a machine designed to produce 1 kg bags of sugar. It is found that the average weight of sugar in the bags is 1.02 kg. Assuming that the weights of the bags are normally distributed, find the standard deviation if 1.7% of the bags weigh below 1 kg. Give your answer correct to the nearest 0.1 gram. Working: Answers: ………………………………………….. (Total 4 marks) 42. When air is released from an inflated balloon it is found that the rate of decrease of the volume of the balloon is proportional to the volume of the balloon. This can be represented by the differential equation dv = – kv, where v is the volume, t is the time and k is the constant of dt proportionality. (a) If the initial volume of the balloon is v0, find an expression, in terms of k, for the volume of the balloon at time t. 37 (b) Find an expression, in terms of k, for the time when the volume is v0 . 2 Working: Answers: (a) ………………………………………….. (b) …………………………………….......... (Total 4 marks) 43. A particle moves along a straight line. When it is a distance s from a fixed point, where s > 1, (3s + 2) the velocity v is given by v = . Find the acceleration when s = 2. (2 s − 1) Working: Answers: ………………………………………….. (Total 4 marks) 38 44. The coordinates of the points P, Q, R and S are (4,1,–1), (3,3,5), (1,0, 2c), and (1,1,2), respectively. (a) Find the value of c so that the vectors OR and PR are orthogonal. For the remainder of the question, use the value of c found in part (a) for the coordinate of the point R. (7) (b) Evaluate PS × PR . (4) (c) Find an equation of the line l which passes through the point Q and is parallel to the vector PR. (3) (d) Find an equation of the plane π which contains the line l and passes through the point S. (4) (e) Find the shortest distance between the point P and the plane π. (4) (Total 22 marks) 45. The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is 6 . 13 If each term of this sequence is positive, and the product of the first term and the third term is 32, find the sum of the first 100 terms of this sequence. (Total 7 marks) 46. Let x and y be real numbers, and ω be one of the complex solutions of the equation z3 = 1. Evaluate: (a) 1 + ω + ω2 (2) (b) (ω x + ω2y) (ω2x + ω y) (4) (Total 6 marks) 39 47. Using mathematical induction, prove that the number 22n – 3n – 1 is divisible by 9, for n...
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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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