answer:First, consider the equation (frac{(x+p)(x+q)(x+5)}{(x+2)^2} = 0). The possible roots are ( -p, -q, ) and ( -5 ). As it has exactly 2 distinct roots, two of these must be the same and different from the third, and none can be ( -2 ) to avoid the denominator becoming zero. Then consider the equation (frac{(x+2p)(x+2)(x+3)}{(x+q)(x+5)} = 0). The possible roots are ( -2p, -2, ) and ( -3 ). Since it has exactly 2 distinct roots, one of the roots must coincide with a value that also makes the denominator zero (either ( -q ) or ( -5 )), and none can be repeated unless it is ( -2 ). Suppose ( -p = -5 ) which gives ( p = 5 ). Then ( -q ) must be ( -2 ) because ( -5 ) and ( -2 ) are the distinct roots of the first equation, so ( q = 2 ). Now, for the second equation to have 2 distinct roots, ( -2p = -10 ) (which equals one of the roots of the denominator, ( -5 )), and ( -2 ) is a valid root. The remaining roots ( -3 ) and ( -5 ) must be distinct and cannot make the denominator zero simultaneously. The conditions are satisfied, as ( -3 neq -5 ) and ( -3 neq -2 ). Thus, both equations work with the stated roots and distinctness: 1. (frac{(x+5)(x+2)(x+5)}{(x+2)^2} = 0) - roots are ( -5, -2 ) (with ( -5 ) repeated) 2. (frac{(x-10)(x+2)(x+3)}{(x+2)(x+5)} = 0) - roots are ( -2, -3 ) Therefore, (100p + q = 100(5) + 2 = boxed{502}).

answer:Since f(x)=ln x+ frac {1}{ln x} (x > 0 and xneq 1), we have f′(x)= frac {1}{x}- frac {1}{x(ln x)^{2}}=0, thus x=e, or x= frac {1}{e}. When xin(0, frac {1}{e}), f′(x) > 0; when xin( frac {1}{e},1) or xin(1,e), f′(x) < 0; when xin(e,+infty), f′(x) > 0. Therefore, x= frac {1}{e} and x=e are respectively the points of local maximum and local minimum of the function f(x), and the function f(x) is monotonically decreasing in ( frac {1}{e},e), hence options A and B are incorrect; When 0 < x < 1, ln x < 0, f(x) < 0, which does not satisfy the inequality, hence option C is incorrect; As long as x_{0}geqslant e, f(x) is an increasing function on (x_{0},+infty), hence option D is correct. Therefore, the correct choice is boxed{D}. By deriving, we can determine that the function is monotonically decreasing on ( frac {1}{e},e), and monotonically increasing on (0, frac {1}{e}) and (e,+infty), which allows us to make a judgment. This question tests the judgment of the truth of propositions and the application of derivative knowledge, where correctly deriving is key.

answer:According to the problem, the equation of the given parabola is y= frac {1}{8}x^2. Therefore, its standard equation is x^2=8y. The focus lies on the y-axis, and p=4. Thus, the coordinates of the focus are (0,2). Therefore, the answer is: (0,2). According to the problem, we first transform the equation of the parabola into its standard form. Analysis reveals that its focus lies on the y-axis, and p=4. Using the formula for the coordinates of the focus of a parabola, we can calculate the answer. This question examines the geometric properties of a parabola, emphasizing the need to first convert the equation of the parabola into its standard form. Hence, the coordinates of the focus are boxed{(0,2)}.

answer:1. **Identify consecutive terms:** The terms given are 25, -50, 100, and -200. 2. **Calculate the ratio of the second term to the first term:** [ r = frac{-50}{25} = -2 ] 3. **Check consistency with other terms:** - The ratio of the third term to the second term is: [ r = frac{100}{-50} = -2 ] - The ratio of the fourth term to the third term is: [ r = frac{-200}{100} = -2 ] Since the ratio is consistent across multiple terms, the common ratio is confirmed. 4. **Conclusion with boxed answer:** [ boxed{-2} ]