Marks 560 a calculator generates a random sequence of

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Unformatted text preview: is zero? (7) A second mass is suspended on another spring. Its distance r cm from the ceiling is modelled by the function r = 60 + 15 cos4πt. The two masses are released at the same instant. (c) Find the value of t when they are first at the same distance below the ceiling. (2) (d) In the first three seconds, how many times are the two masses at the same height? (2) (Total 16 marks) 395 586. It is claimed that the masses of a population of lions are normally distributed with a mean mass of 310 kg and a standard deviation of 30 kg. (a) Calculate the probability that a lion selected at random will have a mass of 350 kg or more. (2) (b) The probability that the mass of a lion lies between a and b is 0.95, where a and b are symmetric about the mean. Find the value of a and of b. (3) (Total 5 marks) 587. Consider the function f given by f(x) = 2 x 2 – 13x + 20 , ( x – 1) 2 x ≠ 1. A part of the graph of f is given below. y 0 x The graph has a vertical asymptote and a horizontal asymptote, as shown. (a) Write down the equation of the vertical asymptote. (1) 396 (b) f(100) = 1.91 f(–100) = 2.09 f(1000) = 1.99 (i) Evaluate f(–1000). (ii) Write down the equation of the horizontal asymptote. (2) (c) Show that f′(x) = 9 x – 27 (x – 1)3 , x ≠ 1. (3) The second derivative is given by f′′(x) = (d) 72 – 18 x , ( x – 1) 4 x ≠ 1. Using values of f′(x) and f′′(x) explain why a minimum must occur at x = 3. (2) (e) There is a point of inflexion on the graph of f. Write down the coordinates of this point. (2) (Total 10 marks) 5 – 2 588. Consider the matrix A = 7 1 . (a) Write down the inverse, A–l. (2) (b) B, C and X are also 2 × 2 matrices. (i) Given that XA + B = C, express X in terms of A–1, B and C. (ii) 6 7 Given that B = 5 – 2 , and C = – 5 0 – 8 7 , find X. (4) (Total 6 marks) 397 589. Consider the points A(1, 2, –4), B(l, 5, 0) and C(6, 5, –12). Find the area of ∆ABC. Working: Answer: ………………………………………….. (Total 6 marks) 398 590. The cumulative frequency curve below indicates the amount of time 250 students spend eating lunch. (a) Estimate the number of students who spend between 20 and 40 minutes eating lunch. (b) If 20 % of the students spend more than x minutes eating lunch, estimate the value of x. 260 240 220 Cumulative frequency 200 180 160 140 120 100 80 60 40 20 0 10 20 30 40 50 60 70 80 Time eating lunch, minutes Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 399 591. The matrices A, B, C and X are all non-singular 3 × 3 matrices. Given that A–lXB = C, express X in terms of the other matrices. Working: Answer: ………………………………………….. (Total 6 marks) 592. A continuous random variable, X, has probability density function f(x) = sin x, 0 ≤ x ≤ π . 2 Find the median of X. Working: Answer: ………………………………………….. (Total 6...
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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.

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