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's the solution to question 8: Given complex numbers Z_1 = 2 + 3i and Z_2 = 2 - i. a) Express (Z_1)/(Z_2) in the form a + bi where a, b R. Step 1: Substitute the given values of Z_1 and Z_2. (Z_1)/(Z_2) = (2 + 3i)/(2 - i) Step 2: Multiply the numerator and denominator by the conjugate of the denominator. The conjugate of 2 - i is 2 + i. (2 + 3i)/(2 - i) × (2 + i)/(2 + i) Step 3: Expand the numerator and the denominator. Numerator: (2 + 3i)(2 + i) = 2(2) + 2(i) + 3i(2) + 3i(i) = 4 + 2i + 6i + 3i^2 Since i^2 = -1: = 4 + 8i - 3 = 1 + 8i Denominator: (2 - i)(2 + i) = 2^2 - i^2 = 4 - (-1) = 4 + 1 = 5 Step 4: Combine the results. (Z_1)/(Z_2) = (1 + 8i)/(5) = (1)/(5) + (8)/(5)i The expression in the form a + bi is (1)/(5) + (8)/(5)i. b) Find the modulus and argument of Z_1 × Z_2. Step 1: Calculate the product Z_1 × Z_2. Z_1 × Z_2 = (2 + 3i)(2 - i) = 2(2) + 2(-i) + 3i(2) + 3i(-i) = 4 - 2i + 6i - 3i^2 Since i^2 = -1: = 4 + 4i - 3(-1) = 4 + 4i + 3 = 7 + 4i Step 2: Find the modulus of Z_1 × Z_2. For a complex number x + yi, the modulus is sqrt(x^2 + y^2). |Z_1 × Z_2| = |7 + 4i| = sqrt(7^2 + 4^2) = sqrt(49 + 16) = sqrt(65) Step 3: Find the argument of Z_1 × Z_2. For a complex number x + yi in the first quadrant, the argument is ^-1((y)/(x)). Since 7 + 4i is in the first quadrant (real part positive, imaginary part positive): (Z_1 × Z_2) = ^-1((4)/(7)) The modulus is sqrt(65) and the argument is ^-1((4)/(7)). c) Given also that Z_3 is a complex number such that |Z_3| = 5 and (Z_3) = ()/(3), express (Z_1 × Z_2)/(Z_3) in polar form. Step 1: From part b), we have Z_1 × Z_2 = 7 + 4i. Its modulus is |Z_1 × Z_2| = sqrt(65). Its argument is (Z_1 × Z_2) = ^-1((4)/(7)). Step 2: We are given Z_3 in polar form: |Z_3| = 5 (Z_3) = ()/(3) Step 3: To find the modulus of the quotient (Z_1 × Z_2)/(Z_3), divide their moduli. |(Z_1 × Z_2)/(Z_3)| = (|Z_1 × Z_2|)/(|Z_3|) = sqrt(65)5 Step 4: To find the argument of the quotient (Z_1 × Z_2)/(Z_3), subtract their arguments. ((Z_1 × Z_2)/(Z_3)) = (Z_1 × Z_2) - (Z_3) = ^-1((4)/(7)) - ()/(3) Step 5: Express the result in polar form, r( + i ). (Z_1 × Z_2)/(Z_3) = sqrt(65)5 ( (^-1((4)/(7)) - ()/(3)) + i (^-1((4)/(7)) - ()/(3)) ) The expression in polar form is sqrt(65)5 ( (^-1((4)/(7)) - ()/(3)) + i (^-1((4)/(7)) - ()/(3)) ).