answer:Prime factorize 252: [ 252 = 2 cdot 126 = 2 cdot 2 cdot 63 = 2 cdot 2 cdot 9 cdot 7 = 2^2 cdot 3^2 cdot 7. ] The distinct prime factors of 252 are 2, 3, and 7. We need to sum these numbers: [ 2 + 3 + 7 = 12. ] Thus, the sum of the distinct prime factors of 252 is boxed{12}.

answer:Let's denote the time it takes Tom Lockheart to shampoo the carpet alone as ( T ) hours. Jason La Rue's rate of work is ( frac{1}{3} ) of the carpet per hour, since he can complete the job in 3 hours. Tom Lockheart's rate of work is ( frac{1}{T} ) of the carpet per hour. When they work together, their combined rate of work is ( frac{1}{3} + frac{1}{T} ) of the carpet per hour. We know that together they take 2 hours to complete the job, so their combined rate of work is ( frac{1}{2} ) of the carpet per hour. Setting the combined rate of work equal to the sum of their individual rates, we get: [ frac{1}{3} + frac{1}{T} = frac{1}{2} ] To solve for ( T ), we first find a common denominator and then solve the equation: [ frac{T + 3}{3T} = frac{1}{2} ] Cross-multiplying to solve for ( T ), we get: [ 2(T + 3) = 3T ] [ 2T + 6 = 3T ] Subtracting ( 2T ) from both sides, we get: [ 6 = T ] So, Tom Lockheart takes boxed{6} hours to shampoo the carpet alone.

answer:Let's analyze the two propositions p and q individually. For the proposition p: the statement that for every x in mathbb{R}, it holds that 2^x > x^2 is false. As a counterexample, consider x=4; we have 2^4 = 16 and 4^2 = 16. Since 16 is not greater than 16, proposition p does not hold for all x in mathbb{R}. For the proposition q: The statement that ab > 1 is a necessary but not sufficient condition for a > 1 and b > 1 is true. If a > 1 and b > 1, then their product ab will indeed be greater than 1. However, it's possible for ab > 1 without both a and b being greater than 1 (for example, if a=2 and b=frac{1}{2}). Thus, while ab > 1 is necessary for both a and b to be greater than 1, it is not sufficient to conclude that both are greater than 1 if only ab > 1 is known. Now, combining the truth values of propositions p and q, we find that p is false and q is true. Therefore, among the choices provided, the true proposition is neg p land q. Thus, the correct answer is: [boxed{B}]

answer:Solution: ((1)) Since (angle AOB = frac{pi}{2}), the distance (d) from point (O) to line (l) is (d = frac{sqrt{2}}{2}r), Thus, (frac{2}{sqrt{k^2 + 1}} = frac{sqrt{2}}{2} cdot sqrt{2} Rightarrow k = pm sqrt{3}). ((2)) Let the distances from the center (O) to the lines (EF) and (GH) be (d_1) and (d_2), respectively, Then (d_1^2 + d_2^2 = |OM|^2 = frac{3}{2}), Therefore, (|EF| = 2sqrt{r^2 - d_1^2} = 2sqrt{2 - d_1^2}), (|GH| = 2sqrt{r^2 - d_2^2} = 2sqrt{2 - d_2^2}), The area (S = frac{1}{2}|EF||GH| = 2sqrt{(2 - d_1^2)(2 - d_2^2)} = sqrt{-4d_2^4 + 6d_2^2 + 4} = sqrt{-4(d_2^2 - frac{3}{4})^2 + frac{25}{4}} leq frac{5}{2}), This equality holds if and only if (d_2^2 = frac{3}{4}), i.e., (d_1 = d_2 = frac{sqrt{3}}{2}), Therefore, the maximum area of the quadrilateral (EGFH) is boxed{frac{5}{2}}.