answer:Solution. Let x (km/h) be the speed of the motorcyclist, x (km/h) be the speed of the car, S (km) be the distance the motorcyclist travels before turning around, then the total length of the track is 2 S+5.25. We have frac{S}{x}=frac{3 S+5.25}{y}, frac{3 x}{8}+6=S, frac{3 y}{8}=S+5.25. This leads to the quadratic equation 4 S^{2}-36 S-63=0, its positive solution is S=10.5, the entire distance traveled by the motorcyclist is 2 S=21.Answer: 21.

answer:1. Find the minimal root of the equation (x^2 - 4x + 2 = 0): The roots of the quadratic equation (ax^2 + bx + c = 0) are given by the quadratic formula: [ x = frac{-b pm sqrt{b^2 - 4ac}}{2a} ] For the equation (x^2 - 4x + 2 = 0), we have (a = 1), (b = -4), and (c = 2). Plugging these values into the quadratic formula, we get: [ x = frac{4 pm sqrt{16 - 8}}{2} = frac{4 pm sqrt{8}}{2} = frac{4 pm 2sqrt{2}}{2} = 2 pm sqrt{2} ] The minimal root is: [ x = 2 - sqrt{2} ]2. Express the sum (x + x^2 + cdots + x^{20}) in a closed form: The sum of a geometric series (S = x + x^2 + cdots + x^{20}) can be written as: [ S = sum_{k=1}^{20} x^k = x frac{x^{20} - 1}{x - 1} ] Substituting (x = 2 - sqrt{2}), we get: [ S = (2 - sqrt{2}) frac{(2 - sqrt{2})^{20} - 1}{(2 - sqrt{2}) - 1} = (2 - sqrt{2}) frac{(2 - sqrt{2})^{20} - 1}{1 - sqrt{2}} ]3. Simplify the expression: Let (A = (2 - sqrt{2})^{20}). Then: [ S = (2 - sqrt{2}) frac{A - 1}{1 - sqrt{2}} ] We need to find the fractional part of (S). First, we approximate (A): [ 2 - sqrt{2} approx 0.5858 ] Since ((2 - sqrt{2})^{20}) is a very small number, we can approximate: [ A approx 0 ] Therefore: [ S approx (2 - sqrt{2}) frac{-1}{1 - sqrt{2}} = (2 - sqrt{2}) frac{-1}{1 - sqrt{2}} = (2 - sqrt{2}) frac{-1}{1 - sqrt{2}} = sqrt{2} - (2 - sqrt{2})^{21} ]4. Estimate the bounds of (S): Since (2 - sqrt{2} < frac{2}{3}), we have: [ (2 - sqrt{2})^{20} < left(frac{2}{3}right)^{20} ] [ left(frac{2}{3}right)^{20} approx 2^{-20} approx 9^{-10} approx 2^{-30} ] Therefore: [ sqrt{2} > S > sqrt{2} - 2^{-9} ] [ sqrt{2} approx 1.414 ] [ 1.414 > S > 1.414 - 0.002 = 1.412 ]5. Determine the fractional part: The fractional part of (S) is: [ {S} = S - lfloor S rfloor ] Since (1.412 < S < 1.414), we have: [ {S} = S - 1 ] [ 0.412 < {S} < 0.414 ]The final answer is (boxed{41}).

answer:To solve this, considering the condition that for the function f(x), forall x in I (where I is the domain of f(x)), when x neq x_{0}, the inequality [f(x)-g(x)](x-x_{0}) > 0 always holds, we use the second derivative set to 0 to solve: f''(x)=- frac {1}{x^{2}}-a=0. It is clear that there is a solution only when a < 0, and this solution is the x-coordinate of the "crossing point".Therefore, x= sqrt { frac {1}{-a}}. According to the problem, x= sqrt { frac {1}{-a}}in(0,e], hence aleqslant - frac {1}{e^{2}}.Thus, the correct choice is: boxed{D}By solving f''(x)=- frac {1}{x^{2}}-a=0, it is clear that there is a solution only when a < 0, and this solution is the x-coordinate of the "crossing point", which leads to the conclusion.This problem mainly tests the new definition, judging the crossing point of a function, and is considered a medium-level question.

answer:The derivative of y=xln x is y'=1+ln x.Thus, the slope of the tangent line at x=e is 1+ln e=2.The point of tangency is (e,e), so the equation of the tangent line at x=e is y-e=2(x-e).Setting x=0 gives y=-e, and setting y=0 gives x=frac{1}{2}e.Therefore, the area of the triangle formed by the tangent line and the two coordinate axes is S=frac{1}{2}cdot ecdotfrac{1}{2}e=frac{e^2}{4}.Hence, the answer is boxed{frac{e^2}{4}}.To solve this problem, we first find the derivative of the given function to determine the slope of the tangent line at x=e. We then use the point-slope form of a linear equation to find the equation of the tangent line. Next, we find the x- and y-intercepts of the tangent line by setting x=0 and y=0, respectively. Finally, we use the formula for the area of a triangle to calculate the desired area. This problem tests our understanding of derivatives and their geometric interpretation, as well as our ability to work with linear equations and calculate the area of a triangle.

answer:【Answer】0【Analysis】Key point: unit digit, sequence patternThe unit digit of each term in the expression cycles through the ten numbers 1, 4, 9, 6, 5, 6, 9, 4, 1, 0. 2012 div 10=201 cdots 2,The unit digit of the expression is equal to the unit digit of 201 times(1+4+6+9+5+9+6+4+1)+1+4, which is 0.

answer:To solve this problem, we start by noting that there are 10 basic events when drawing 2 balls out of 5, which are: {1,2}, {1,3}, {1,4}, {1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, {4,5}. (1) Let event A be "drawing two balls of the same color". Event A includes 4 basic events, which are: {1,2}, {1,3}, {2,3}, {4,5}. Therefore, the probability of event A happening is P = frac{4}{10} = frac{2}{5}. So, the probability that the two drawn balls are of the same color is boxed{frac{2}{5}}. (2) Let event B be "drawing two balls of different colors, and at least one of the balls has an odd number". Event B includes 5 basic events, which are: {1,4}, {1,5}, {2,5}, {3,4}, {3,5}. Therefore, the probability of event B happening is P(B) = frac{5}{10} = frac{1}{2}. So, the probability that the two drawn balls are of different colors, and at least one of the balls has an odd number is boxed{frac{1}{2}}.