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Unformatted text preview: the triangle is equilateral. Working: Answer: …………………………………………........ (Total 6 marks) ˆ 665. The diagram shows a sector AOB of a circle of radius 1 and centre O, where AOB = θ. The lines (AB1), (A1B2), (A2B3) are perpendicular to OB. A1B1, A2B2 are all arcs of circles with centre O. A A1 A2 O B3 B2 B1 B 448 Calculate the sum to infinity of the arc lengths AB + A1B1 + A2B2 + A3B3 + … Working: Answer: …………………………………………........ (Total 6 marks) 666. (a) The point P(1, 2, 11) lies in the plane π1. The vector 3i – 4 j + k is perpendicular to π1. Find the Cartesian equation of π1. (2) (b) The plane π2 has equation x + 3y – z = –4. (i) Show that the point P also lies in the plane π2. (ii) Find a vector equation of the line of intersection of π1 and π2. (5) (c) Find the acute angle between π1 and π2. (5) (Total 12 marks) 667. Jack and Jill play a game, by throwing a die in turn. If the die shows a 1, 2, 3 or 4, the player who threw the die wins the game. If the die shows a 5 or 6, the other player has the next throw. Jack plays first and the game continues until there is a winner. (a) Write down the probability that Jack wins on his first throw. (1) 449 (b) Calculate the probability that Jill wins on her first throw. (2) (c) Calculate the probability that Jack wins the game. (3) (Total 6 marks) 668. Let f(x) be the probability density function for a random variable X, where kx 2 , for 0 ≤ x ≤ 2 f ( x) 0, otherwise (a) Show that k = 3 . 8 (2) (b) Calculate (i) E(X); (ii) the median of X. (6) (Total 8 marks) 669. A complex number z is such that (a) z = z − 3i . Show that the imaginary part of z is 3 . 2 (2) 450 (b) Let z1 and z2 be the two possible values of z, such that z = 3. (i) Sketch a diagram to show the points which represent z1 and z2 in the complex plane, where z1 is in the first quadrant. (ii) Show that arg z1 = (iii) Find arg z2. π . 6 (4) (c) z1k z 2 2i = π, find a value of k. Given that arg (4) (Total 10 marks) 670. Find an expression for the sum of the first 35 terms of the series ln x2 + ln giving your answer in the form ln x2 x2 x2 + ln 2 + ln 3 + … y y y xm , where m, n ∈ yn . (Total 5 marks) 671. The temperature T °C of an object in a room, after t minutes, satisfies the differential equation dT = k(T – 22), where k is a constant. dt (a) Solve this equation to show that T = Aekt + 22, where A is a constant. (3) 451 (b) When t = 0, T = 100, and when t = 15, T = 70. (i) Use this information to find the value of A and of k. (ii) Hence find the value of t when T = 40. (7) (Total 10 marks) 672. (a) Show that cos(A + B) + cos(A – B) = 2 cosAcosB (2) (b) Let Tn(x) = cos(n arccosx) where x is a real number, x ∈ [–1, 1] and n is a positive integer. (i) Find T1(x). (ii) Show that T2(x) = 2x2 – 1. (5) (c) (i) Use the result in part (a) to show that Tn+1(x) + Tn–1(x) = 2xTn(x). (ii) Hence or otherwise, prove by induction that Tn(x) is a polynomial of degree n. (12) (...
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