This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
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Problem 89: Step 1: Expand the left side of the equation. We are given the equation (1+2i)(x+yi) = 4+3i. Multiply the terms on the left side: (1)(x) + (1)(yi) + (2i)(x) + (2i)(yi) = 4+3i x + yi + 2xi + 2yi^2 = 4+3i Since i^2 = -1, substitute this into the equation: x + yi + 2xi - 2y = 4+3i Step 2: Group the real and imaginary parts on the left side. (x - 2y) + (y + 2x)i = 4+3i Step 3: Equate the real parts and the imaginary parts of both sides. For two complex numbers to be equal, their real parts must be equal, and their imaginary parts must be equal. Equating real parts: x - 2y = 4 (Equation 1) Equating imaginary parts: 2x + y = 3 (Equation 2) Step 4: Solve the system of linear equations. From Equation 2, we can express y in terms of x: y = 3 - 2x Substitute this expression for y into Equation 1: x - 2(3 - 2x) = 4 x - 6 + 4x = 4 5x - 6 = 4 Add 6 to both sides: 5x = 10 Divide by 5: x = 2 Now substitute x=2 back into the expression for y: y = 3 - 2(2) y = 3 - 4 y = -1 So, x=2 and y=-1. Step 5: Compare with the given options. The values x=2 and y=-1 match option A. The final answer is A. x=2, y=-1. Problem 90: Step 1: Rewrite the equation. We need to find the four distinct roots of the equation z^4 = 1. This means we are looking for the fourth roots of unity. Step 2: Express 1 in polar form. The complex number 1 can be written as 1 + 0i. Its modulus is r = sqrt(1^2 + 0^2) = 1. Its argument is = ((0)/(1)) = 0. In polar form, 1 = 1((0) + i(0)). To find all roots, we use the general form of the argument: 1 = (0 + 2k) + i(0 + 2k), where k is an integer. Step 3: Apply De Moivre's Theorem for roots. The n-th roots of a complex number w = r( + i) are given by: z_k = [n]r((( + 2k)/(n)) + i(( + 2k)/(n))) For z^4 = 1, we have n=4, r=1, and =0. z_k = [4]1(((0 + 2k)/(4)) + i((0 + 2k)/(4))) z_k = ((k)/(2)) + i((k)/(2)) We need to find the roots for k=0, 1, 2, 3. Step 4: Calculate each root. For k=0: z_0 = ((0)/(2)) + i((0)/(2)) = (0) + i(0) = 1 + i(0) = 1 For k=1: z_1 = ((1)/(2)) + i((1)/(2)) = (()/(2)) + i(()/(2)) = 0 + i(1) = i For k=2: z_2 = ((2)/(2)) + i((2)/(2)) = () + i() = -1 + i(0) = -1 For k=3: z_3 = ((3)/(2)) + i((3)/(2)) = 0 + i(-1) = -i The four distinct roots are 1, i, -1, -i. Step 5: Compare with the given options. The roots 1, -1, i, -i match option C. The final answer is C. (1, -1, i, -i).