With scale factor k a horizontal translation of p

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Unformatted text preview: ine (i) the percentage of cars travelling at a speed in excess of 105 km h–1; (ii) the speed which is exceeded by 15% of the cars. (4) (Total 11 marks) 245 353. The diagram below shows part of the graph of the function f : x a – x 3 + 2 x 2 + 15 x . y 40 35 30 25 20 15 10 5 A –3 –2 –1 –5 –10 –15 P –20 Q B 1 2 3 4 5 x The graph intercepts the x-axis at A(–3,0), B(5,0) and the origin, O. There is a minimum point at P and a maximum point at Q. (a) The function may also be written in the form f : x a – x( x – a ) ( x – b), where a < b . Write down the value of (i) a; (ii) b. (2) (b) Find (i) f ′(x); (ii) the exact values of x at which f '(x) = 0; (iii) the value of the function at Q. (7) 246 (c) (i) Find the equation of the tangent to the graph of f at O. (ii) This tangent cuts the graph of f at another point. Give the x-coordinate of this point. (4) (d) Determine the area of the shaded region. (2) (Total 15 marks) 354. The diagram below shows the positions of towns O, A, B and X. diagram not to scale X O B A Town A is 240 km East and 70 km North of O. Town B is 480 km East and 250 km North of O. Town X is 339 km East and 238 km North of O. An airplane flies at a constant speed of 300 km h –1 from O towards A. (a) (i) 0.96 Show that a unit vector in the direction of OA is 0.28 . (ii) v1 Write down the velocity vector for the airplane in the form . v 2 (iii) How long does it take for the airplane to reach A? (5) 247 At A the airplane changes direction so it now flies towards B . The angle between the original direction and the new direction is θ as shown in the following diagram. This diagram also shows the point Y, between A and B , where the airplane comes closest to X. diagram not to scale X O (b) B A Use the scalar product of two vectors to find the value of θ in degrees. (4) (c) (i) Write down the vector AX . (ii) – 3 Show that the vector n = is perpendicular to AB . 4 (iii) By finding the projection of AX in the direction of n, calculate the distance XY. (6) (d) How far is the airplane from A when it reaches Y ? (3) (Total 18 marks) 355. A ball is dropped vertically from a great height. Its velocity v is given by v = 50 – 50e–0.2t, t ≥ 0 where v is in metres per second and t is in seconds. (a) Find the value of v when (i) t = 0; (ii) t = 10. (2) 248 (b) (i) Find an expression for the acceleration, a, as a function of t. (ii) What is the value of a when t = 0? (3) (c) (i) As t becomes large, what value does v approach? (ii) As t becomes large, what value does a approach? (iii) Explain the relationship between the answers to parts (i) and (ii). (3) (d) Let y metres be the distance fallen after t seconds. (i) Show that y = 50t + 250e–0.2t + k, where k is a constant. (ii) Given that y = 0 when t = 0, find the value of k. (iii) Find the time required to fall 250 m, giving your answer correct to four significant figures. (7) (Total 15 marks) 356. Bags of cement are labelled 25 kg. The bags are filled by machine and the actual weights are...
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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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