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Unformatted text preview: …………………........ (Total 6 marks) 477 1 − 1 2 708. Given that the matrix A = 2 p 3 is singular, find the value of p. 1 − 2 5 Working: Answer: …………………………………………........ (Total 6 marks) 709. The sum of the first n terms of a series is given by Sn = 2n2 – n, where n ∈ + . (a) Find the first three terms of the series. (b) Find an expression for the nth term of the series, giving your answer in terms of n. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks) 478 710. Given that (a + i)(2 – bi) = 7 – i, find the value of a and of b, where a, b ∈ . Working: Answer: …………………………………………........ (Total 6 marks) 711. If y = ln(2x – 1), find d2 y . dx 2 Working: Answer: …………………………………………........ (Total 6 marks) 479 712. A fair six-sided die, with sides numbered 1, 1, 2, 3, 4, 5 is thrown. Find the mean and variance of the score. Working: Answer: …………………………………………........ (Total 6 marks) 713. (a) Find the largest set S of values of x such that the function f(x) = 1 3 − x2 takes real values. (b) Find the range of the function f defined on the domain S. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks) 480 714. (a) (b) Find the expansion of (2 + x)5, giving your answer in ascending powers of x. By letting x = 0.01 or otherwise, find the exact value of 2.015. Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks) 715. The diagram below shows a circle centre O and radius OA = 5 cm. The angle AOB = 135°. A B O 481 Find the area of the shaded region. Working: Answer: …………………………………………........ (Total 6 marks) 716. Consider the equation e–x = cos2x, for 0 ≤ x ≤ 2π. (a) How many solutions are there to this equation? (b) Find the solution closest to 2π, giving your answer to four decimal places. Working: Answers: (a) ……………………………………….. (b) ………………………………………….. (Total 6 marks) 482 717. Consider the four points A(1, 4, –1), B(2, 5, –2), C(5, 6, 3) and D(8, 8, 4). Find the point of intersection of the lines (AB) and (CD). Working: Answer: …………………………………………........ (Total 6 marks) 718. A continuous random variable X has probability density function given by f(x) = k (2x – x2), f(x) = 0, (a) Find the value of k. (b) for 0 ≤ x ≤ 2 elsewhere. Find P(0.25 ≤ x ≤ 0.5). Working: Answers: (a) ………………………………………….. (b) ………………………………………….. (Total 6 marks) 483 719. Given that z ∈ , solve the equation z3 – 8i = 0, giving your answers in the form z = r (cosθ + i...
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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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