# 3 marks 138 197 for the function f x a x2 1n x x

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Unformatted text preview: Find the x-coordinate of P, the point of intersection of the two curves. (2) (c) If the tangents to the curves at P meet the y-axis at Q and R, calculate the area of the triangle PQR. (6) 148 (d) Prove that the two tangents at the points where x = a, a > 0, on each curve are always perpendicular. (4) (Total 14 marks) 215. A uniform rod of length l metres is placed with its ends on two supports A, B at the same horizontal level. l metres A B y(x) x metres If y (x) metres is the amount of sag (ie the distance below [AB]) at a distance x metres from support A, then it is known that ( ) d2 y 1 = x 2 – lx . 2 3 dx 125l ( ) 1 x 3 − lx 2 + 1 . Show that dz = 1 x 2 − lx . dx 125l 3 2 1500 125l 3 3 (i) Let z = (ii) Given that dw = z and w(0) = 0, find w(x). dx (iii) (a) 2 Show that w satisfies d w = 1 3 (x2 – lx), and that w(l) = w(0) = 0. dx 2 125l (8) (b) Find the sag at the centre of a rod of length 2.4 metres. (2) (Total 10 marks) 149 216. A satellite relies on solar cells for its power and will operate provided that at least one of the cells is working. Cells fail independently of each other, and the probability that an individual cell fails within one year is 0.8. (a) For a satellite with ten solar cells, find the probability that all ten cells fail within one year. (2) (b) For a satellite with ten solar cells, find the probability that the satellite is still operating at the end of one year. (2) (c) For a satellite with n solar cells, write down the probability that the satellite is still operating at the end of one year. Hence, find the smallest number of solar cells required so that the probability of the satellite still operating at the end of one year is at least 0.95. (5) (Total 9 marks) 217. The lifetime of a particular component of a solar cell is Y years, where Y is a continuous random variable with probability density function 0 when y < 0 f ( y) = - y/2 when y ≥ 0. 0.5e (a) Find the probability, correct to four significant figures, that a given component fails within six months. Each solar cell has three components which work independently and the cell will continue to run if at least two of the components continue to work. (3) (b) Find the probability that a solar cell fails within six months. (4) (Total 7 marks) 218. (a) Given matrices A, B, C for which AB = C and det A ≠ 0, express B in terms of A and C. (2) 150 (b) 1 Let A = 2 3 2 −1 −3 3 2 , D = 2 (i) − 7 − 4 and C = 5 5 7. 10 Find the matrix DA; (ii) − 4 13 7 − 2 3 −9 Find B if AB = C. (3) (c) Find the coordinates of the point of intersection of the planes x + 2y + 3z = 5, 2x – y + 2z = 7 and 3x – 3j + 2z = 10. (2) (Total 7 marks) 219. (a) If u = i +2j + 3k and v = 2i – j + 2k, show that u × v = 7i + 4j – 5k. (2) (b) Let w = λu + µv where λ and µ are scalars. Show that w is perpendicular to the line of intersection of the planes x + 2y + 3z = 5 and 2x – y + 2z = 7 for all values of λ and µ. (4) (Total 6 marks) 220. (a) Let y = a + b sin 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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