Arc acb b the area of the shaded region working

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Unformatted text preview: + x 3 ) 2 12 x 2 x 3 – 1 . (1 + x 3 ) 3 Find the x-coordinates of the points of inflexion of the graph of f. (6) (c) The table below gives some values of f(x) and 2f(x). x f(x) 2f(x) 1 2 1.4 0.534188 1.068376 1.8 0.292740 0.585480 2.2 0.171703 0.343407 2.6 0.107666 0.215332 3 (i) 1 0.071429 0.142857 Use the trapezium rule with five sub-intervals to approximate the integral ∫ f (x )dx. 3 1 ∫ f (x )dx = 0.637599, use a diagram to explain why your answer is 3 (ii) Given that 1 greater than this. (5) (Total 15 marks) 283 414. Consider the arithmetic series 2 + 5 + 8 +.... (a) Find an expression for Sn, the sum of the first n terms. (b) Find the value of n for which Sn = 1365. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 415. A particle is projected along a straight line path. After t seconds, its velocity v metres per 1 second is given by v = . 2 + t2 (a) Find the distance travelled in the first second. (b) Find an expression for the acceleration at time t. Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 284 416. (a) (b) Express the complex number 8i in polar form. The cube root of 8i which lies in the first quadrant is denoted by z. Express z (i) in polar form; (ii) in cartesian form. Working: Answers: (a) ………………………………………….. (b) (i) ........................................................... (ii) ……………………………………... (Total 6 marks) 285 417. The matrix A is given by 2 1 k A = 1 k – 1 3 4 2 Find the values of k for which A is singular. Working: Answers: ……………………………………………….. (Total 6 marks) 418. Find the angle between the vectors v = i + j + 2k and w = 2i + 3j + k. Give your answer in radians. Working: Answer: ……………………………………………….. (Total 6 marks) 286 419. (a) Use integration by parts to find ∫x 2 ln x dx. 2 (b) Evaluate ∫x 2 lnxdx. 1 Working: Answers: (a) ………………………………………….. (b) .................................................................. (Total 6 marks) 420. The probability that it rains during a summer’s day in a certain town is 0.2. In this town, the probability that the daily maximum temperature exceeds 25 °C is 0.3 when it rains and 0.6 when it does not rain. Given that the maximum daily temperature exceeded 25 °C on a particular summer’s day, find the probability that it rained on that day. Working: Answer: ……………………………………………….. (Total 6 marks) 287 421. The vector equations of the lines L1 and L2 are given by L1: r = i + j + k + λ(i + 2j + 3k); L2: r = i + 4j + 5k + µ(2i + j + 2k). The two lines intersect at the point P. Find the position vector of P. Working: Answer: ………………………………………...
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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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