answer:**Analysis** This problem tests the multiplication and division operations of complex numbers in algebraic form and examines the basic concepts of complex numbers. It is a fundamental problem. By utilizing the multiplication and division rules of complex numbers, simplify frac{1-ai}{1+i}. Then, combine this with the concept of complex numbers to obtain the answer. **Step-by-Step Solution** 1. Simplify the given complex number z by multiplying both the numerator and denominator by the conjugate of the denominator: z = frac{1 - ai}{1 + i} cdot frac{1 - i}{1 - i} = frac{(1 - ai)(1 - i)}{(1 + i)(1 - i)} 2. Apply the distributive property and the fact that i^2 = -1: z = frac{1 - a - i + ai^2}{1 - i + i - i^2} = frac{1 - a - i - a}{1 + 1} = frac{1 - a}{2} - frac{1 + a}{2}i 3. According to the problem, the imaginary part of z is -3. Therefore, we can set up the equation: - frac{1 + a}{2} = -3 4. Solve for a by first multiplying both sides by -2 to get rid of the fraction and the negative sign: 1 + a = 6 5. Subtract 1 from both sides: a = 5 Thus, the value of a is boxed{5}.

answer:Given that a number is congruent to 1 modulo 23, we can write it as (23n + 1). To find the largest five-digit negative integer of this form, we need to ensure (23n + 1 < -9999). Solving this inequality for (n): [ 23n + 1 < -9999 implies 23n < -10000 implies n < -10000/23 approx -434.7826 ] The largest integer less than -434.7826 is -435. We substitute this value back into the equation for (n): [ 23(-435) + 1 = -9995 + 1 = -9994 ] So, the largest five-digit negative integer congruent to 1 modulo 23 is (boxed{-9994}).

answer:Let's denote the width of the river at the starting point as W (in yards). Since the river's width increases uniformly by 2 yards every 10 meters along its length, we can express the width of the river at any point as a function of the distance x (in meters) from the starting point: Width at distance x = W + (2 yards/10 meters) * x Rodney and Todd row along the river at a rate of 5 m/s for 30 seconds, so the distance they cover is: Distance = Speed * Time = 5 m/s * 30 s = 150 meters At this point, the width of the river is given as 80 yards. Therefore, we can set up the equation: 80 yards = W + (2 yards/10 meters) * 150 meters Now, we solve for W: 80 yards = W + (2 yards/10 meters) * (150 meters) 80 yards = W + (2 yards/meter) * 15 80 yards = W + 30 yards Subtract 30 yards from both sides to find W: W = 80 yards - 30 yards W = 50 yards So, the width of the river at the starting point is boxed{50} yards.

answer:**Analysis** This question mainly examines the rules of addition and subtraction of two vectors and their geometric meanings, belonging to medium difficulty. Since |overrightarrow{a} + overrightarrow{b}| and |overrightarrow{a} - overrightarrow{b}| represent the lengths of the two diagonals of the parallelogram with overrightarrow{a} and overrightarrow{b} as adjacent sides, and from |overrightarrow{a} + overrightarrow{b}| = |overrightarrow{a} - overrightarrow{b}|, it can be concluded that the diagonals of this parallelogram are equal, hence the parallelogram is a rectangle. Therefore, we can draw the conclusion. **Solution** From the rules of addition and subtraction of two vectors and their geometric meanings, |overrightarrow{a} + overrightarrow{b}| and |overrightarrow{a} - overrightarrow{b}| represent the lengths of the two diagonals of the parallelogram with overrightarrow{a} and overrightarrow{b} as adjacent sides. Given |overrightarrow{a} + overrightarrow{b}| = |overrightarrow{a} - overrightarrow{b}|, it can be concluded that the diagonals of this parallelogram are equal, hence the parallelogram is a rectangle, which implies overrightarrow{a} perp overrightarrow{b}. Therefore, the correct choice is boxed{text{A}}.