6 marks 505 750 a sum of 5 000 is invested at a

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Unformatted text preview: field ABC. One side of the triangle, [AC], is 104 m, a second side, [AB], is 65 m and the angle between these two sides is 60°. (a) Use the cosine rule to calculate the length of the third side of the field. (3) (b) Given that sin 60° = 3 , find the area of the field in the form p 3 where p is an integer. 2 (3) 514 Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts A1 and A2 by constructing a straight fence [AD] of length x metres, as shown on the diagram below. C 104 m A2 A 30° D x 30° A1 65 m B 65x . 4 (i) Show that the area of Al is given by (ii) Find a similar expression for the area of A2. (iii) (c) Hence, find the value of x in the form q 3 , where q is an integer. (7) (d) (i) ˆ ˆ Explain why sin ADC = sin ADB . (ii) Use the result of part (i) and the sine rule to show that BD 5 =. DC 8 (5) (Total 18 marks) 515 759. The diagram below shows the graphs of f(x) = 1 + e2x, g(x) = 10x + 2, 0 ≤ x ≤ 1.5. y f g 16 12 p 8 4 0.5 (a) 1 1.5 x (i) Write down an expression for the vertical distance p between the graphs of f and g. (ii) Given that p has a maximum value for 0 ≤ x ≤ 1.5, find the value of x at which this occurs. (6) The graph of y = f(x) only is shown in the diagram below. When x = a, y = 5. y 16 12 8 5 4 0.5 a (b) (i) 1.5 x Find f–1(x). (ii) 1 Hence show that a = ln 2. (5) 516 (c) The region shaded in the diagram is rotated through 360° about the x-axis. Write down an expression for the volume obtained. (3) (Total 14 marks) 760. Consider the function h : x a x–2 ( x – 1) 2 , x ≠ 1. A sketch of part of the graph of h is given below. A y P x Not to scale B The line (AB) is a vertical asymptote. The point P is a point of inflexion. (a) Write down the equation of the vertical asymptote. (1) (b) Find h'(x), writing your answer in the form a–x ( x – 1) n where a and n are constants to be determined. (4) 517 (c) Given that h ′′ ( x) = 2x – 8 ( x – 1) 4 , calculate the coordinates of P. (3) (Total 8 marks) 761. Bag A contains 2 red balls and 3 green balls. Two balls are chosen at random from the bag without replacement. Let X denote the number of red balls chosen. The following table shows the probability distribution for X X 1 2 P(X = x) (a) 0 3 10 6 10 1 10 Calculate E(X), the mean number of red balls chosen. (3) Bag B contains 4 red balls and 2 green balls. Two balls are chosen at random from bag B. (b) (i) Draw a tree diagram to represent the above information, including the probability of each event. (ii) Hence find the probability distribution for Y, where Y is the number of red balls chosen. (8) 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. (5) (d) Given that two red balls are obtained, find the conditional probability that a 1 or 6 was rolled on the die. (3) (Total 19 marks) 518 762. In this question, distance is in kilometers, time is in hours. A balloon is moving at a constant height with a speed of l8 km h–1, in the 3 direction of the vector 4 0 At time t = 0, the balloon is at point B...
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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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