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(Total 3 marks) 184 263. The diameters of discs produced by a machine are normally distributed with a mean of 10 cm
and standard deviation of 0.1 cm. Find the probability of the machine producing a disc with a
diameter smaller than 9.8 cm.
Working: Answers:
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(Total 3 marks) 264. The equation of motion of a particle with mass m, subjected to a force kx can be written as
dv
kx = mv , where x is the displacement and v is the velocity. When x = 0, v = v0. dx Find v, in
dx
terms of v0, k and m, when x = 2.
Working: Answers:
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(Total 3 marks) 185 265. Solve, for x, the equation log2 (5x2 – x – 2) = 2 + 2 log2 x.
Working: Answers:
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(Total 3 marks) a 266. Find the value of a such that ∫ cos 2 x dx = 0.740. Give your answer to 3 decimal places. 0 Working: Answers:
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(Total 3 marks) 186 267. Find the xcoordinate, between –2 and 0, of the point of inflexion on the graph of the function
f : x a x 2 e x . Give your answer to 3 decimal places.
Working: Answers:
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(Total 3 marks) 268. Find the area of the region enclosed by the graphs of y = sin x and y = x2 – 2x + 1.5, where
0 ≤ x ≤ π.
Working: Answers:
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(Total 3 marks) 187 269. The diagram shows a sketch of the graph of y = f′(x) for a ≤ x ≤ b.
y y = f’(x) a b x On the grid below, which has the same scale on the xaxis, draw a sketch of the graph of
y = f(x) for a ≤ x ≤ b, given that f(0) = 0 and f(x) ≥ 0 for all x. On your graph you should clearly
indicate any minimum or maximum points, or points of inflexion. y a b x Working: (Total 3 marks) 188 270. For the vectors a = 2i + j – 2k, b = 2i –j – k and c = i + 2j + 2k, show that:
(a) a × b = –3i – 2j – 4k
(2) (b) (a × b) × c = –(b • c)a
(3)
(Total 5 marks) 271. Three points A, B and C have coordinates (2, 1, –2), (2, –1, –1) and (1, 2, 2) respectively. The
vectors OA , OB and OC , where O is the origin, form three concurrent edges of a
parallelepiped OAPBCQSR as shown in the following diagram.
P S A Q
B O (a) R
C Find the coordinates of P, Q, R and S.
(4) (b) Find an equation for the plane OAPB.
(2) (c) Calculate the volume, V, of the parallelepiped given that
V = OA × OB • OC (2)
(Total 8 marks) 189 272. (a) 2 2 Sketch and label the graphs of f ( x ) = e – x and g ( x ) = e x – 1 for
0 ≤ x ≤ 1, and shade the region A which is bounded by the graphs and the yaxis.
(3) (b) Let the xcoordinate of the point of intersection of the curves y = f(x) and y = g(x) be p.
Without finding the value of p, show that p
< area of region A < p.
2
(4) (c) Find the value of p correct to four decimal places.
(2) (d) Express the area of region A as a definite integral and calculate its value.
(3)
(Total 12 marks) 2...
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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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