answer:Let's denote the total number of valid votes as V. Since there were 83 invalid votes, the total number of valid votes is: V = Total polled votes - Invalid votes V = 90,083 - 83 V = 90,000 Let's denote the number of votes the defeated candidate got as D and the number of votes the winning candidate got as W. We know that the winning candidate got 9,000 more votes than the defeated candidate, so: W = D + 9,000 Since W and D are the only two candidates, their votes must add up to the total number of valid votes: W + D = V (D + 9,000) + D = 90,000 2D + 9,000 = 90,000 2D = 90,000 - 9,000 2D = 81,000 D = 81,000 / 2 D = 40,500 So, the defeated candidate got 40,500 votes. To find the percentage of votes the defeated candidate got, we divide the number of votes they got by the total number of valid votes and multiply by 100: Percentage of votes for the defeated candidate = (D / V) * 100 Percentage of votes for the defeated candidate = (40,500 / 90,000) * 100 Percentage of votes for the defeated candidate = 0.45 * 100 Percentage of votes for the defeated candidate = 45% Therefore, the defeated candidate got boxed{45%} of the votes.

answer:f(x) is a decreasing function on (-infty, 0). Proof: Assume x_1, x_2 in (-infty, 0) and x_1 < x_2, then we have -x_1 > -x_2 > 0... (4 points) Since f(x) is a decreasing function on [0, +infty), we have f(-x_1) < f(-x_2)... (7 points) Furthermore, since f(x) is an odd function over mathbb{R}, we have -f(x_1) < -f(x_2), which implies f(x_1) > f(x_2). (10 points) Therefore, f(x) is a decreasing function on (-infty, 0). (12 points) Thus, the final answer is f(x) is a decreasing function on (-infty, 0), which can be encapsulated as boxed{text{f(x) is a decreasing function on } (-infty, 0)}.

answer:The domain of the function f(x) = ln x + 2x - 8 is (1, +infty). We evaluate the function at some specific points: At x = 1, f(1) = ln 1 + 2 cdot 1 - 8 = 0 + 2 - 8 = -6 < 0, At x = 2, f(2) = ln 2 + 2 cdot 2 - 8 = ln 2 + 4 - 8 = ln 2 - 4 < 0, At x = 3, f(3) = ln 3 + 2 cdot 3 - 8 = ln 3 + 6 - 8 = ln 3 - 2 < 0, At x = 4, f(4) = ln 4 + 2 cdot 4 - 8 = ln 4 + 8 - 8 = ln 4 > 0, Since f(3) < 0 and f(4) > 0, we can conclude that there is a sign change in the function values between x = 3 and x = 4. According to the Intermediate Value Theorem which ensures that a continuous function has at least one root in an interval where it changes sign, we can deduce that the function f(x) has a root in the interval (3, 4). Therefore, the correct answer is boxed{text{C}}, and the function has a zero within the interval (3,4).

answer:Since cos left( frac {pi}{4}-alpha right)= frac { sqrt {2}}{4}, Therefore, sin 2alpha=cos left( frac {pi}{2}-2alpha right)=cos 2left( frac {pi}{4}-alpha right) =2cos ^{2}left( frac {pi}{4}-alpha right)-1=2times left( frac { sqrt {2}}{4} right)^{2}-1=- frac {3}{4}. Hence, the answer is - frac {3}{4}. This problem involves simplifying trigonometric functions and applying the double-angle cosine formula, which is a basic question. The final answer is boxed{-frac{3}{4}}.