The median of x 5 total 12 marks 788 use mathematical

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Unformatted text preview: that ABC is 85°. (i) Find the distance from A to C. (ii) Find the area of the region ABC with the fence in this position. (5) (c) To form the second region, he moves the fencing so that point C is closer to point A. Find the new distance from A to C. (4) (d) Find the minimum length of fence [BC] needed to enclose a triangular region ABC. (2) (Total 14 marks) 827. Let f(x) = 1 sin2x + cos x for 0 ≤ x ≤ 2π. 2 (a) (i) Find f′(x). One way of writing f′(x) is –2 sin2x – sinx + 1. (ii) Factorize 2sin2x + sinx – 1. (iii) Hence or otherwise, solve f′(x) = 0. (6) The graph of y = f(x) is shown below. y A b 0a 2π x B There is a maximum point at A and a minimum point at B. 566 (b) Write down the x-coordinate of point A. (1) (c) The region bounded by the graph, the x-axis and the lines x = a and x = b is shaded in the diagram above. (i) Write down an expression that represents the area of this shaded region. (ii) Calculate the area of this shaded region. (5) (Total 12 marks) 1 828. In this question the vector represents a displacement of 1 km east, 0 0 and the vector represents a displacement of 1 km north. 1 The diagram below shows the positions of towns A, B and C in relation to an airport O, which is at the point (0, 0). An aircraft flies over the three towns at a constant speed of 250 km h–1. y B O x A C Town A is 600 km west and 200 km south of the airport. Town B is 200 km east and 400 km north of the airport. Town C is 1200 km east and 350 km south of the airport. 567 (a) (i) Find AB . (ii) 0.8 Show that the vector of length one unit in the direction of AB is . 0 .6 (4) An aircraft flies over town A at 12:00, heading towards town B at 250 km h–1. p Let be the velocity vector of the aircraft. Let t be the number of hours in flight after 12:00. q The position of the aircraft can be given by the vector equation x − 600 p = y − 200 + t q . (i) 200 Show that the velocity vector is 150 . (ii) Find the position of the aircraft at 13:00. (iii) (b) At what time is the aircraft flying over town B? (6) Over town B the aircraft changes direction so it now flies towards town C. It takes five hours to travel the 1250 km between B and C. Over town A the pilot noted that she had 17 000 litres of fuel left. The aircraft uses 1800 litres of fuel per hour when travelling at 250 km h–1. When the fuel gets below 1000 litres a warning light comes on. (c) How far from town C will the aircraft be when the warning light comes on? (7) (Total 17 marks) 568 829. Residents of a small town have savings which are normally distributed with a mean of $ 3 000 and a standard deviation of $ 500. (i) What percentage of townspeople have savings greater than $ 3 200? (ii) Two townspeople are chosen at random. What is the probability that both of them have savings between $ 2 300 and $ 3 300? (iii) The percentage of townspeople with savings less than d dollars is 74.22 %. Find the value of d. (Total 8 marks) 2 830. Let f(x) = 3 x . 5x − 1 (a) Write down the equati...
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