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Unformatted text preview: , and z2 = 1 – i. 2 Write z1 and z2 in the form r(cos θ + i sin θ), where r > 0 and – π π ≤θ≤ . 2 2 (6) (b) Show that z1 = cos π + i sin π . z2 12 12 (2) (c) Find the value of z1 in the form a + bi, where a and b are to be determined exactly in z2 radical (surd) form. Hence or otherwise find the exact values of cos π and sin π . 12 12 (4) (Total 12 marks) 107. Consider the points A(l, 2, 1), B(0, –1, 2), C(1, 0, 2), and D(2, –1, –6). (a) Find the vectors AB and BC . (2) (b) Calculate AB × BC . (3) 73 (c) Hence, or otherwise find the area of triangle ABC. (2) (d) Find the equation of the plane P containing the points A, B, and C. (3) (e) Find a set of parametric equations for the line through the point D and perpendicular to the plane P. (2) (f) Find the distance from the point D to the plane P. (3) (g) Find a unit vector which is perpendicular to the plane P. (2) (h) The point E is a reflection of D in the plane P. Find the coordinates of E. (4) (Total 21 marks) x 1n x − kx, x > 0 108. Consider the function fk(x) = , where k ∈ x=0 0, (a) Find the derivative of fk(x), x > 0. (2) (b) Find the interval over which f(x) is increasing. The graph of the function fk(x) is shown below. (2) y 0 A x 74 (i) Show that the stationary point of fk(x) is at x = ek–1. (ii) (c) One x-intercept is at (0, 0).Find the coordinates of the other x-intercept. (4) (d) Find the area enclosed by the curve and the x-axis. (5) (e) Find the equation of the tangent to the curve at A. (2) (f) Show that the area of the triangular region created by the tangent and the coordinate axes is twice the area enclosed by the curve and the x-axis. (2) (g) Show that the x-intercepts of fk(x) for consecutive values of k form a geometric sequence. (3) (Total 20 marks) 109. The continuous random variable X has probability density function f (x) where e − ke kx , f k( x) = 0, (a) 0 ≤ x ≤1 otherwise Show that k = 1. (3) (b) What is the probability that the random variable X has a value that lies between 1 and 1 ? Give your answer in terms of e. 4 2 (2) (c) Find the mean and variance of the distribution. Give your answers exactly, in terms of e. (6) The random variable X above represents the lifetime, in years, of a certain type of battery. (d) Find the probability that a battery lasts more than six months. (2) 75 A calculator is fitted with three of these batteries. Each battery fails independently of the other two. Find the probability that at the end of six months (e) none of the batteries has failed; (2) (f) exactly one of the batteries has failed. (2) (Total 17 marks) 110. Consider the set U = {1, 3, 5, 9, 11, 13} under the operation *, where * is multiplication modulo 14. (In all parts of this problem, the general properties of multiplication modulo n may be assumed.) (a) Show that (3 * 9) * 13 = 3 * (9 * 13). (2) (b) Show that (U, *) is a group. (11) (c) (i) Define a cyclic group. (2) (ii) Show that (U, *) is cyclic and find all its generators. (7)...
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