answer:To translate the parabola y=-left(x-1right)^{2} one unit to the right and two units up, we follow these steps:1. One Unit to the Right: When we translate a function one unit to the right, we subtract one from the x variable inside the function. Therefore, translating y=-left(x-1right)^{2} one unit to the right gives us:[y = -left((x-1) - 1right)^{2}]Simplifying the expression inside the parentheses:[y = -left(x - 2right)^{2}]2. Two Units Up: To translate the function two units up, we add two to the entire function. Applying this to our current equation gives us:[y = -left(x - 2right)^{2} + 2]Thus, after translating the parabola one unit to the right and two units up, we get the equation:[y = -left(x - 2right)^{2} + 2]Therefore, the correct answer is boxed{text{B}}.

answer:Let the total number of products in the factory for models A, B, and C be 3k, 4k, and 7k respectively, where k is a common multiplier. According to the stratified sampling method, the sample should maintain the same ratio of 3:4:7.Since there are 9 pieces of model A within the sample, we can set up the proportion for model A as follows: frac{9}{n} = frac{3k}{3k + 4k + 7k} Solving for n we get: frac{9}{n} = frac{3k}{14k} 9 cdot 14k = 3k cdot n 126k = 3k cdot n n = frac{126k}{3k} n = 42 Hence, the size of the sample n is boxed{42}.

answer:Given the equation x-3y=4, we want to find the value of left(x-3yright)^{2}+2x-6y-10.First, we simplify the expression by recognizing that 2x-6y can be rewritten as 2(x-3y):begin{align*}left(x-3yright)^{2}+2x-6y-10 &= left(x-3yright)^{2}+2left(x-3yright)-10 &= left(x-3yright)^{2}+2cdot4-10 quad text{(since x-3y=4)} &= 4^{2}+2cdot4-10 &= 16+8-10 &= 24-10 &= 14.end{align*}Therefore, the value of the given expression is boxed{14}.

answer:Solution: (a=log_{2}frac{1}{4} 1), (c=left(frac{4}{5}right)^{2} in (0,1)),Therefore, (a < c < b),Hence, the correct option is: boxed{B}.This can be determined by using the monotonicity of functions.This question tests the understanding of the monotonicity of functions, reasoning, and computational skills, and is considered a basic question.

answer:Analysis:To prove this statement, we need to demonstrate two aspects:1. If ax^2+bx+c=0 has a root of 1, then a+b+c=0.2. If a+b+c=0, then ax^2+bx+c=0 has a root of 1.Proof:1. Necessary Condition: Assume that 1 is a root of the equation ax^2+bx+c=0. Substituting x=1 into the equation, we get a(1)^2+b(1)+c=a+b+c=0. Therefore, if the equation has a root of 1, then it is necessary that a+b+c=0.2. Sufficient Condition: Assume that a+b+c=0. We need to prove that 1 is a root of the equation ax^2+bx+c=0. Substituting x=1 into the equation, we get a(1)^2+b(1)+c=a+b+c. Since a+b+c=0, substituting this into the equation gives 0=0, which is true. Therefore, if a+b+c=0, it is sufficient for the equation to have a root of 1.In conclusion, the necessary and sufficient condition for the equation ax^2+bx+c=0 to have a root of 1 is boxed{a+b+c=0}.

answer:Solve 16 pairs.We point out that if there are a rooks in a row, then there are a-1 pairs of rooks that can attack each other in that row. Therefore, the number of rooks that can attack each other horizontally will not be less than 8 pairs. Similarly, the number of rooks that can attack each other vertically will not be less than 8 pairs.There exists an arrangement where exactly 16 pairs of rooks can attack each other, for example, by placing all the rooks on the two main diagonals (top-left to bottom-right, top-right to bottom-left).