answer:To analyze each option step by step: **For option A:** Given overrightarrow{a} perp overrightarrow{b}, we know that overrightarrow{a} cdot overrightarrow{b} = 0. Thus, we have: [ overrightarrow{a} cdot overrightarrow{b} = (1,-2) cdot (-1,m) = 1 cdot (-1) + (-2) cdot m = -1 - 2m = 0. ] Solving for m gives: [ -1 - 2m = 0 Rightarrow -2m = 1 Rightarrow m = -frac{1}{2}. ] Therefore, option A is incorrect because it states m = -1. **For option B:** Given overrightarrow{a} parallel overrightarrow{b}, we have the ratio of the corresponding components equal, which implies: [ frac{1}{-1} = frac{-2}{m} Rightarrow m = 2. ] Substituting m = 2 into overrightarrow{b} gives overrightarrow{b} = (-1, 2). Thus, the dot product overrightarrow{a} cdot overrightarrow{b} is: [ overrightarrow{a} cdot overrightarrow{b} = 1 cdot (-1) + (-2) cdot 2 = -1 - 4 = -5. ] Hence, option B is correct. **For option C:** Given m = 1, we substitute into overrightarrow{b} to get overrightarrow{b} = (-1, 1). The difference overrightarrow{a} - overrightarrow{b} is: [ overrightarrow{a} - overrightarrow{b} = (1, -2) - (-1, 1) = (1 + 1, -2 - 1) = (2, -3). ] The magnitude of overrightarrow{a} - overrightarrow{b} is: [ |overrightarrow{a} - overrightarrow{b}| = sqrt{2^2 + (-3)^2} = sqrt{4 + 9} = sqrt{13}. ] Thus, option C is correct. **For option D:** Given m = -2, we substitute into overrightarrow{b} to get overrightarrow{b} = (-1, -2). The cosine of the angle between overrightarrow{a} and overrightarrow{b} is given by: [ cos<overrightarrow{a}, overrightarrow{b}> = frac{overrightarrow{a} cdot overrightarrow{b}}{|overrightarrow{a}| |overrightarrow{b}|} = frac{1 cdot (-1) + (-2) cdot (-2)}{sqrt{1^2 + (-2)^2} cdot sqrt{(-1)^2 + (-2)^2}} = frac{-1 + 4}{sqrt{1 + 4} cdot sqrt{1 + 4}} = frac{3}{5}. ] This does not correspond to an angle of 60^{circ}, so option D is incorrect. Therefore, the correct options are boxed{text{BC}}.

answer:First, we simplify sqrt[3]{1728}. We start by finding the prime factorization of 1728: 1728 = 2^6 cdot 3^3. We can then write sqrt[3]{1728} = sqrt[3]{2^6 cdot 3^3} = sqrt[3]{2^6} cdot sqrt[3]{3^3}. This simplifies further, as the cube root of 2^6 is 2^2 = 4 and the cube root of 3^3 is 3. Therefore, sqrt[3]{1728} = 4 cdot 3 = 12. Thus, we have a = 12 and b = 1 (since there's no radical part left), giving us a + b = 12 + 1 = boxed{13}.

answer:To find the constant ( k ), we need to expand the right side of the equation and then compare the coefficients of the corresponding terms on both sides of the equation. The given equation is: [ -x^2 - (k + 12)x - 8 = -(x - 2)(x - 4) ] First, let's expand the right side: [ -(x - 2)(x - 4) = -[x^2 - 4x - 2x + 8] ] [ -(x - 2)(x - 4) = -[x^2 - 6x + 8] ] [ -(x - 2)(x - 4) = -x^2 + 6x - 8 ] Now, let's compare the coefficients of the corresponding terms on both sides of the equation: The coefficient of ( x^2 ) on the left side is ( -1 ), and on the right side, it is also ( -1 ), so they match. The coefficient of ( x ) on the left side is ( -(k + 12) ), and on the right side, it is ( 6 ). So we have: [ -(k + 12) = 6 ] Now, let's solve for ( k ): [ -k - 12 = 6 ] [ -k = 6 + 12 ] [ -k = 18 ] [ k = -18 ] Therefore, the constant ( k ) is ( boxed{-18} ).

answer:To solve this, when x = -frac{b}{2a} = -frac{-2(2k-1)}{2 times 1} = 2k-1, the function takes its minimum value. The minimum value m = frac{4ac-b^2}{4a} = frac{4 times (3k^2 - 2k + 6) - [-2(2k-1)]^2}{4} = -k^2 + 2k + 5 = -(k-1)^2 + 6. When k = 1, m reaches its maximum value, which is 6. At this point, x = 2k-1 = 2 times 1 - 1 = 1. Therefore, the answer is: boxed{1}.