6 marks 812 the following diagram shows a circle of

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Unformatted text preview: …………………….......... (Total 6 marks) 851. The function f is defined for x > 2 by f(x) = ln x + ln (x – 2) – ln (x2 – 4). (a) Express f(x) in the form ln x . x+a (b) Find an expression for f–1(x). Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 582 852. Let y = log3 z, where z is a function of x. The diagram shows the straight line L, which represents the graph of y against x. y L –2 (a) (b) –1 1 2 3 4 5 6 7 x Using the graph or otherwise, estimate the value of x when z = 9. The line L passes through the point 1, log 3 5 . Its gradient is 2. Find an expression for z 9 in terms of x. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 583 853. The function f is defined by f(x) = epx(x + 1), here p ∈ . (i) Show that f′(x) = epx(p(x + 1) + 1). (ii) (a) Let f(n)(x) denote the result of differentiating f(x) with respect to x, n times. Use mathematical induction to prove that f(n)(x) = pn–1epx (p(x + 1) + n), n ∈ + . (7) (b) When p = 3 , there is a minimum point and a point of inflexion on the graph of f. Find the exact value of the x-coordinate of (i) the minimum point; (ii) the point of inflexion. (4) (c) Let p = 1 . Let R be the region enclosed by the curve, the x-axis and the lines x = –2 and 2 x = 2. Find the area of R. (2) (Total 13 marks) 854. (a) 2 − 2 1 The plane π1 has equation r = 1 + λ 1 + µ − 3 . 1 8 − 9 2 1 1 The plane π2 has the equation r = 0 + s 2 + t 1 . 1 1 1 (i) For points which lie in π1 and π2, show that, λ = µ. (ii) Hence, or otherwise, find a vector equation of the line of intersection of π and π2. (5) (b) y The plane π3 contains the line 2 − x = = z + 1 and is perpendicular to 3i – 2j + k. 3 −4 Find the cartesian equation of π3. (4) 584 (c) Find the intersection of π1, π2 and π3. (3) (Total 12 marks) 855. A company buys 44 % of its stock of bolts from manufacturer A and the rest from manufacturer B. The diameters of the bolts produced by each manufacturer follow a normal distribution with a standard deviation of 0.16 mm. The mean diameter of the bolts produced by manufacturer A is 1.56 mm. 24.2 % of the bolts produced by manufacturer B have a diameter less than 1.52 mm. (a) Find the mean diameter of the bolts produced by manufacturer B. (3) A bolt is chosen at random from the company’s stock. (b) Show that the probability that the diameter is less than 1.52 mm is 0.312, to three significant figures. (4) (c) The diameter of the bolt is found to be less than 1.52 mm. Find the probability that the bolt was produced by manufacturer B. (3) (d) Manufacturer B makes 8000 bolts in one day. It makes a profit of $ 1.50 on each bolt sold, on condition that its diameter measures between 1.52 mm and 1.83 mm. Bolts whose diameters measure less than 1.52 mm must b...
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