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Unformatted text preview: f ( x) = cos OPA = x 2 – 8 x + 40
√ {( x 2 – 16 x + 80) ( x 2 + 100)} , 0 ≤ x ≤ 15. 210 (e) Consider the equation f (x) = 1.
(i) Explain, in terms of the position of the points O, A, and P, why this
equation has a solution. (ii) Find the exact solution to the equation.
(5)
(Total 16 marks) 301. Intelligence Quotient (IQ) in a certain population is normally distributed with a mean of 100
and a standard deviation of 15.
(a) What percentage of the population has an IQ between 90 and 125?
(2) (b) If two persons are chosen at random from the population, what is the probability that
both have an IQ greater than 125?
(3) (c) The mean IQ of a random group of 25 persons suffering from a certain brain disorder
was found to be 95.2. Is this sufﬁcient evidence, at the 0.05 level of signiﬁcance, that
people suffering from the disorder have, on average, a lower IQ than the entire
population? State your null hypothesis and your alternative hypothesis, and explain your
reasoning.
(4)
(Total 9 marks) 302. The function f is given by (a) (i) Show that f ( x) = f ′ ( x) = 1n 2 x
,
x 1 – 1n 2 x
x2 x > 0. . Hence
(ii) prove that the graph of f can have only one local maximum or minimum point; (iii) ﬁnd the coordinates of the maximum point on the graph of f.
(6) 211 (b) 21n 2 x – 3
or otherwise,
x3
ﬁnd the coordinates of the point of inﬂexion on the graph of f.
By showing that the second derivative f ′′ ( x) = (6) (c) The region S is enclosed by the graph of f , the xaxis, and the vertical line
through the maximum point of f , as shown in the diagram below.
y
y=f(x) 0
(i) x Would the trapezium rule overestimate or underestimate the area of S? Justify your
answer by drawing a diagram or otherwise.
(3) (ii) Find ∫ f ( x) dx , by using the substitution u = ln 2x, or otherwise.
(4) (iii) Using ∫ f ( x) dx , ﬁnd the area of S.
(4) (d) The NewtonRaphson method is to be used to solve the equation f (x) = 0.
(i) Show that it is not possible to ﬁnd a solution using a starting value of
x 1 = 1.
(3) (ii) Starting with x1 = 0.4, calculate successive approximations x2, x3, ...
for the root of the equation until the absolute error is less than 0.01.
Give all answers correct to ﬁve decimal places.
(4)
(Total 30 marks) 212 1
1 303. Let f(t) = t 3 1 – 5 . Find 2t 3 ∫ f (t ) dt. Working: Answers:
…………………………………………..
(Total 3 marks) 304. Solve 2 sin x = tan x, where – π
2 <x< π
2 . Working: Answers:
…………………………………………..
(Total 3 marks) 213 305. Find the gradient of the tangent to the curve 3x2 + 4y2 = 7 at the point where x = 1 and y > 0.
Working: Answers:
…………………………………………..
(Total 3 marks) 306. Let f : x a 1
– 2 . Find
x2 (a) the set of real values of x for which f is real and finite; (b) the range of f. Working: Answers:
(a) …………………………………………..
(b) .........................................................
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 Fall '13
 Apple
 The Land

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