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Unformatted text preview: …………………………………….. (b) .................................................................. (Total 6 marks) 605. Let f(x) = x+4 x–2 , x ≠ –1 and g(x) = , x ≠ 4. x–4 x +1 Find the set of values of x such that f(x) ≤ g(x). Working: Answer: ………………………………………….. (Total 6 marks) 409 606. A committee of four children is chosen from eight children. The two oldest children cannot both be chosen. Find the number of ways the committee may be chosen. Working: Answer: ………………………………………….. (Total 6 marks) 607. Solve 2(5x+1) = 1 + 3 , giving the answer in the form a + log5 b, where a, b ∈ 5x . Working: Answer: ………………………………………….. (Total 6 marks) 410 608. An airplane is flying at a constant speed at a constant altitude of 3 km in a straight line that will take it directly over an observer at ground level. At a given instant the observer notes that the 1 1 angle θ is π radians and is increasing at radians per second. Find the speed, in kilometres 3 60 per hour, at which the airplane is moving towards the observer. Airplane x 3 km Observer Working: Answer: ………………………………………….. (Total 6 marks) 609. The point A (2, 5, –1) is on the line L, which is perpendicular to the plane with equation x + y + z – 1 = 0. (a) Find the Cartesian equation of the line L. (2) (b) Find the point of intersection of the line L and the plane. (4) 411 (c) The point A is reflected in the plane. Find the coordinates of the image of A. (2) (d) Calculate the distance from the point B(2, 0, 6) to the line L. (4) (Total 12 marks) 610. (a) Use mathematical induction to prove De Moivre’s theorem (cosθ + i sinθ)n = cos(nθ) + i sin(nθ), n ∈ + . (7) (b) Consider z5 – 32 = 0. (i) 2π 2π Show that z1 = 2 cos + i sin is one of the complex roots of this 5 5 equation. (ii) Find z12, z13, z14, z15, giving your answer in the modulus argument form. (iii) Plot the points that represent z1, z12, z13, z14 and z15, in the complex plane. (iv) The point z1n is mapped to z1n+1 by a composition of two linear transformations, where n = 1, 2, 3, 4. Give a full geometric description of the two transformations. (9) (Total 16 marks) 611. (a) Express 3 cosθ – sinθ in the form r cos(θ + α), where r > 0 and 0 < α < π , giving r 2 and α as exact values. (3) (b) Hence, or otherwise, for 0 ≤ θ ≤ 2π, find the range of values of 3 cosθ – sinθ. (2) 412 (c) Solve 3cosθ – sinθ = –l, for 0 ≤ θ ≤ 2π, giving your answers as exact values. (5) (Total 10 marks) 612. Prove that π π sin 4θ (1 – cos 2θ ) = tan θ, for 0 < θ < , and θ ≠ . cos 2θ (1 – cos 4θ ) 2 4 (Total 5 marks) 613. Use the substitution y = xv to show that the general solution to the differential equation, dy = 0, x > 0 i s (x2 + y2) + 2xy dx x3 + 3xy2 = k, where k is a constant. (Total 6 marks) 614. A curve has equation f(x) = (a) Show that f″ (x) =...
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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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