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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Here are the solutions for Question 3. 3A. Given Z_1 = 2+4i and Z_2 = 1+i. i) Product Z_1 Z_2 Step 1: Multiply the complex numbers. Z_1 Z_2 = (2+4i)(1+i) Step 2: Expand the product. (2+4i)(1+i) = 2(1) + 2(i) + 4i(1) + 4i(i) = 2 + 2i + 4i + 4i^2 Step 3: Simplify using i^2 = -1. 2 + 6i + 4(-1) = 2 + 6i - 4 = -2 + 6i The product is -2 + 6i. ii) Division Z_1 / Z_2 Step 1: Write the division as a fraction. (Z_1)/(Z_2) = (2+4i)/(1+i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator, which is 1-i. (2+4i)/(1+i) = ((2+4i)(1-i))/((1+i)(1-i)) Step 3: Expand the numerator and denominator. Numerator: (2+4i)(1-i) = 2(1) + 2(-i) + 4i(1) + 4i(-i) = 2 - 2i + 4i - 4i^2 = 2 + 2i - 4(-1) = 2 + 2i + 4 = 6 + 2i Denominator: (1+i)(1-i) = 1^2 - i^2 = 1 - (-1) = 1 + 1 = 2 Step 4: Combine the results and simplify. (6+2i)/(2) = (6)/(2) + (2i)/(2) = 3 + i The division is 3 + i. 3B. Given Z_1 = 1-2i and Z_2 = 1+2i. i) Product Z_1 Z_2 Step 1: Multiply the complex numbers. Z_1 Z_2 = (1-2i)(1+2i) Step 2: Use the difference of squares formula (a-b)(a+b) = a^2 - b^2. 1^2 - (2i)^2 Step 3: Simplify using i^2 = -1. 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5 The product is 5. ii) Division Z_1 / Z_2 Step 1: Write the division as a fraction. (Z_1)/(Z_2) = (1-2i)/(1+2i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator, which is 1-2i. (1-2i)/(1+2i) = ((1-2i)(1-2i))/((1+2i)(1-2i)) Step 3: Expand the numerator and denominator. Numerator: (1-2i)(1-2i) = 1(1) + 1(-2i) - 2i(1) - 2i(-2i) = 1 - 2i - 2i + 4i^2 = 1 - 4i + 4(-1) = 1 - 4i - 4 = -3 - 4i Denominator: (1+2i)(1-2i) = 1^2 - (2i)^2 = 1 - 4i^2 = 1 - 4(-1) = 1 + 4 = 5 Step 4: Combine the results and simplify. (-3-4i)/(5) = -(3)/(5) - (4)/(5)i The division is -(3)/(5) - (4)/(5)i.