# 9 marks 335 504 let x be a set containing n elements

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Unformatted text preview: company manufactures television sets. They claim that the lifetime of a set is normally distributed with a mean of 80 months and standard deviation of 8 months. (a) What proportion of television sets break down in less than 72 months? (2) (b) (i) Calculate the proportion of sets which have a lifetime between 72 months and 90 months. (ii) Illustrate this proportion by appropriate shading in a sketch of a normal distribution curve. (5) (c) If a set breaks down in less than x months, the company replace it free of charge. They replace 4% of the sets. Find the value of x. (3) (Total 10 marks) 531. Consider the function f(x) = cos x + sin x. (a) π ) = 0. 4 (i) Show that f(– (ii) Find in terms of π, the smallest positive value of x which satisfies f(x) = 0. (3) 355 The diagram shows the graph of y = ex (cos x + sin x), – 2 ≤ x ≤ 3. The graph has a maximum turning point at C(a, b) and a point of inflexion at D. y 6 C(a, b) 4 D 2 –2 (b) Find –1 1 2 3 x dy . dx (3) (c) Find the exact value of a and of b. (4) π (d) Show that at D, y = 2e 4 . (5) (e) Find the area of the shaded region. (2) (Total 17 marks) 356 532. The matrices A, B, X are given by 3 1 A= – 5 6 , B = 4 8 0 – 3 , X = a b c d , where a, b, c, d ∈ . Given that AX + X = Β, find the exact values of a, b, c and d. (Total 8 marks) 533. A geometric sequence has all positive terms. The sum of the first two terms is 15 and the sum to infinity is 27. Find the value of (a) the common ratio; (b) the first term. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 357 534. Find all the values of θ in the interval [0, π] which satisfy the equation cos2θ = sin2θ. Working: Answers: …....………………………………………….. …….................................................................. (Total 6 marks) 535. Given that a = i + 2j – k, b = –3i + 2j + 2k and c = 2i– 3j + 4k, find (a × b) ⋅ c. Working: Answer: ………………………………………….. (Total 6 marks) 358 536. The polynomial x3 + ax2 – 3x + b is divisible by (x – 2) and has a remainder 6 when divided by (x + 1). Find the value of a and of b. Working: Answers: …....………………………………………….. …….................................................................. (Total 6 marks) 3 – 2 537. Given that A = – 3 4 and I = matrix. 1 0 0 1 , find the values of λ for which (A – λI) is a singular Working: Answers: …....………………………………………….. …….................................................................. (Total 6 marks) 359 538. When a boy plays a game at a fair, the probability that he wins a prize is 0.25. He plays the game 10 times. Let X denote the total number of prizes that he wins. Assuming that the games are independent, find (a) E(X) (b) P (X ≤ 2). Working: Answers: (a) ………………………………………….. (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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