4 marks 200 290 find the angle between the following

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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 x-axis, 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 Newton-Raphson 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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