x a x 1 x 1 a determine the inverse function

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Unformatted text preview: ve at 13:00 with velocity vector km h–1. 12 (e) Show that the position of the Boadicea for t ≥ 1 is given by x 13 5 = +t y – 8 12 (2) 171 (f) Find how far apart the two ships are at 15:00. (4) (Total 20 marks) 248. A formula for the depth d metres of water in a harbour at a time t hours after midnight is π d = P + Q cos t , 0 ≤ t ≤ 24, 6 where P and Q are positive constants. In the following graph the point (6, 8.2) is a minimum point and the point (12, 14.6) is a maximum point. d 15 (12, 14.6) 10. (6, 8.2) 5 0 (a) 6 12 18 24 t Find the value of (i) Q; (ii) P. (3) (b) Find the first time in the 24-hour period when the depth of the water is 10 metres. (3) (c) (i) Use the symmetry of the graph to find the next time when the depth of the water is 10 metres. (ii) Hence find the time intervals in the 24-hour period during which the water is less than 10 metres deep. (4) 172 249. A ball is thrown vertically upwards into the air. The height, h metres, of the ball above the ground after t seconds is given by h = 2 + 20t – 5t2, t ≥ 0 (a) Find the initial height above the ground of the ball (that is, its height at the instant when it is released). (2) (b) Show that the height of the ball after one second is 17 metres. (2) (c) At a later time the ball is again at a height of 17 metres. (i) Write down an equation that t must satisfy when the ball is at a height of 17 metres. (ii) Solve the equation algebraically. (4) (d) dh . dt (i) Find (ii) Find the initial velocity of the ball (that is, its velocity at the instant when it is released). (iii) Find when the ball reaches its maximum height. (iv) Find the maximum height of the ball. (7) (Total 15 marks) 250. The function f is given by f ( x) =1 – 2x 1+ x 2 173 (i) To display the graph of y = f(x) for –10 ≤ x ≤ 10, a suitable interval for y, a ≤ y ≤ b must be chosen. Suggest appropriate values for a and b . (ii) (a) Give the equation of the asymptote of the graph. (3) (b) Show that f ′ ( x) = 2x 2 – 2 (1 + x 2 ) 2 . (4) (c) Use your answer to part (b) to find the coordinates of the maximum point of the graph. (3) (d) (i) Either by inspection or by using an appropriate substitution, find ∫ f ( x) dx (ii) Hence find the exact area of the region enclosed by the graph of f, the x-axis and the y-axis. (8) (Total 18 marks) 174 251. Find the sum to infinity of the geometric series – 12 + 8 – 16 + .... 3 Working: Answers: ………………………………………….. (Total 3 marks) 252. Find the values of a and b, where a and b are real, given that (a + bi)(2 – i) = 5 – i. Working: Answers: ………………………………………….. (Total 3 marks) 175 253. The diagram shows a sketch of part of the graph of f(x) = x2 and a sketch of part of the graph of g(x) = –x2 + 6x – 13 y y=f(x) x y=g(x) (a) Write down the coordinates of the maximum point of y = g(x). 176 The graph of y = g(x) can be obtained from the graph of y = f(x) by first reflecting the graph of y = f(x...
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