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 to questions 1a and 1b. 1a. Find the modulus and argument of For a complex number Z = x + yi: The modulus is |Z| = sqrt(x^2 + y^2). The argument is (Z) = , where = (y)/(x), and is chosen based on the quadrant of (x, y). i) Z = 4 + 3i Step 1: Identify the real and imaginary parts. Here, x = 4 and y = 3. Step 2: Calculate the modulus. |Z| = sqrt(4^2 + 3^2) = sqrt(16 + 9) = sqrt(25) = 5 Step 3: Calculate the argument. = (3)/(4) Since x > 0 and y > 0, is in the first quadrant. = ((3)/(4)) ≈ 36.87^ or 0.6435 radians The modulus is 5 and the argument is ((3)/(4)). ii) Z = 1 + i Step 1: Identify the real and imaginary parts. Here, x = 1 and y = 1. Step 2: Calculate the modulus. |Z| = sqrt(1^2 + 1^2) = sqrt(1 + 1) = sqrt(2) Step 3: Calculate the argument. = (1)/(1) = 1 Since x > 0 and y > 0, is in the first quadrant. = (1) = ()/(4) or 45^ The modulus is sqrt(2) and the argument is ()/(4). iii) Z = + i Step 1: Identify the real and imaginary parts. Here, x = and y = . Step 2: Calculate the modulus. |Z| = sqrt(()^2 + ()^2) = sqrt(^2 + ^2) Using the identity ^2 + ^2 = 1: |Z| = sqrt(1) = 1 Step 3: Calculate the argument. The standard polar form of a complex number is r( + i). We have Z = + i. Using the trigonometric identities = (()/(2) - ) and = (()/(2) - ): Z = (()/(2) - ) + i(()/(2) - ) Comparing this to the polar form, the argument is ()/(2) - . The modulus is 1 and the argument is ()/(2) - . 1b. Evaluate The powers of i follow a cycle: i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1. This cycle repeats every 4 powers. To evaluate i^n, find the remainder of n when divided by 4. i) i^20 Step 1: Divide the exponent by 4. 20 ÷ 4 = 5 with a remainder of 0 Step 2: Use the remainder to find the value. Since the remainder is 0, i^20 is equivalent to i^4. i^20 = i^4 = 1 The value is 1. ii) Z = -1 + i^2 Step 1: Substitute the value of i^2. We know i^2 = -1. Z = -1 + (-1) Step 2: Simplify the expression. Z = -1 - 1 = -2 The value is -2. iii) i^-4 Step 1: Rewrite the expression with a positive exponent. i^-4 = (1)/(i^4) Step 2: Substitute the value of i^4. We know i^4 = 1. (1)/(i^4) = (1)/(1) = 1 The value is 1. iv) i^5 Step 1: Divide the exponent by 4. 5 ÷ 4 = 1 with a remainder of 1 Step 2: Use the remainder to find the value. Since the remainder is 1, i^5 is equivalent to i^1. i^5 = i^1 = i The value is i.