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Unformatted text preview: = – 1 3 2 1 +t 4 – 1 Working: Answer: ………………………………………….. (Total 6 marks) 309 459. The diagram shows part of the curve y = sin x. The shaded region is bounded by the curve and 3π the lines y = 0 and x = . 4 y 3π 4 Given that sin π x 3π 3π 2 2 = and cos =– , calculate the exact area of the shaded region. 2 2 4 4 Working: Answer: ………………………………………….. (Total 6 marks) 310 460. $1000 is invested at 15% per annum interest, compounded monthly. Calculate the minimum number of months required for the value of the investment to exceed $3000. Working: Answer: ………………………………………….. (Total 6 marks) 461. Consider the trigonometric equation 2 sin2 x = 1 + cos x. (a) Write this equation in the form f(x) = 0, where f(x) = a cos2 x + b cos x + c, and a, b, c ∈ . 311 (b) Factorize f(x). (c) Solve f(x) = 0 for 0° ≤ x ≤ 360°. Working: Answers: (a) ………………………………………….. (b) .................................................................. (c) ………………………………………….. (Total 6 marks) 312 462. The sketch shows part of the graph of y = f(x) which passes through the points A(–1, 3), B(0, 2), C(l, 0), D(2, 1) and E(3, 5). 8 7 6 E 5 4 A 3 B 2 D 1 C –4 –3 –2 –1 0 1 2 3 4 5 –1 –2 A second function is defined by g(x) = 2f(x – 1). (a) Calculate g(0), g(1), g(2) and g(3). (b) On the same axes, sketch the graph of the function g(x). Working: Answers: (a) ………………………………………….. .................................................................. (Total 6 marks) 313 463. In a suburb of a large city, 100 houses were sold in a three-month period. The following cumulative frequency table shows the distribution of selling prices (in thousands of dollars). Selling price P ($ 1000) P ≤ 100 P ≤ 200 P ≤ 300 P ≤ 400 P ≤ 500 Total number of houses 12 58 87 94 100 (a) Represent this information on a cumulative frequency curve, using a scale of 1 cm to represent $ 50000 on the horizontal axis and 1 cm to represent 5 houses on the vertical axis. (4) (b) Use your curve to find the interquartile range. (3) The information above is represented in the following frequency distribution. Selling price P ($ 1000) Number of houses (c) 0 < P ≤ 100 100 < P ≤ 200 200 < P ≤ 300 300 < P ≤ 400 400 < P ≤ 500 12 46 29 a b Find the value of a and of b. (2) (d) Use mid-interval values to calculate an estimate for the mean selling price. (2) (e) Houses which sell for more than $ 350 000 are described as De Luxe. (i) Use your graph to estimate the number of De Luxe houses sold. Give your answer to the nearest integer. (ii) Two De Luxe houses are selected at random. Find the probability that both have a selling price of more than $ 400 000. (4) (Total 15 marks) 314 464. The diagram shows a square ABCD of side 4 cm. The midpoints P, Q, R, S of the sides are joined to form a second s...
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