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Unformatted text preview: ..................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... ............................................................................................................................................... (Total 6 marks) 783. The function f is defined on the domain x ≥ 1 by f(x) = (a) ln x . x (i) Show, by considering the first and second derivatives of f, that there is one f maximum point on the graph of f. (ii) State the exact coordinates of this point. (9) (iii) The graph of f has a point of inflexion at P. Find the x-coordinate of P. (3) 539 Let R be the region enclosed by the graph of f, the x-axis and the line x = 5. (c) Find the exact value of the area of R. (6) (d) The region R is rotated through an angle 2n about the x-axis. Find the volume of the solid of revolution generated. (3) (Total 21 marks) 784. A farmer owns a triangular field ABC. The side [AC] is 104 m, the side [AB] is 65 m and the angle between these two sides is 60°. (a) Calculate the length of the third side of the field. (3) (b) Find the area of the field in the form p 3 , where p is an integer. (3) Let D be a point on [BC] such that [AD] bisects the 60° angle. The farmer divides the field into two parts by constructing a straight fence [AD] of length x metres. (c) (i) Show that the area of the smaller part is given by 65x and find an expression for 4 the area of the larger part. (ii) Hence, find the value of x in the form q 3 , where q is an integer. (8) (d) Prove that BD 5 =. DC 8 (6) (Total 20 marks) 540 785. (a) x−2 y −2 z −3 x−2 y −3 z −4 and intersect and find the = = = = 1 3 1 1 4 2 coordinates of P, the point of intersection. Show that lines (8) (b) Find the Cartesian equation of the plane ∏ that contains the two lines. (6) (c) The point q (3, 4, 3) lies on ∏. The line L passes through the midpoint of [PQ]. Point S is on L such that PS = QS = 3 , and the triangle PQS is normal to the plane ∏. Given that there are two possible pos...
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