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(Total 6 marks) 851. The function f is defined for x > 2 by f(x) = ln x + ln (x â€“ 2) â€“ ln (x2 â€“ 4).
(a) Express f(x) in the form ln x . x+a (b) Find an expression for fâ€“1(x). Working: Answers:
(a) â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..
(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 â€“2 (a)
(b) â€“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) â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦â€¦..
(b) ..................................................................
(Total 6 marks) 583 853. The function f is defined by f(x) = epx(x + 1), here p âˆˆ . (i) Show that fâ€²(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) = pnâ€“1epx (p(x + 1) + n), n âˆˆ + .
(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 xcoordinate 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 xaxis and the lines x = â€“2 and
2
x = 2. Find the area of R.
(2)
(Total 13 marks) 854. (a) 2 âˆ’ 2
1 The plane Ï€1 has equation r = 1 + Î» 1 + Âµ âˆ’ 3 .
1
8 âˆ’ 9 2 1 1 The plane Ï€2 has the equation r = 0 + s 2 + t 1 . 1 1 1 (i) For points which lie in Ï€1 and Ï€2, show that, Î» = Âµ. (ii) Hence, or otherwise, find a vector equation of the line of intersection of Ï€ and Ï€2.
(5) (b) y
The plane Ï€3 contains the line 2 âˆ’ x =
= z + 1 and is perpendicular to 3i â€“ 2j + k.
3
âˆ’4 Find the cartesian equation of Ï€3.
(4) 584 (c) Find the intersection of Ï€1, Ï€2 and Ï€3.
(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 companyâ€™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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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.
 Fall '13
 Apple
 The Land

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