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.
ScanSolve AI Answer
Here's the solution to the problems. Problem 2: Express (2+j3)/(j(4-j5)) + (2)/(j) in the form a+jb. Step 1: Simplify the denominator of the first term and the second term. For the first term's denominator: j(4-j5) = j4 - j^2 5 = j4 - (-1)5 = 5+j4 For the second term: (2)/(j) = (2)/(j) × (-j)/(-j) = (-j2)/(-j^2) = (-j2)/(-(-1)) = (-j2)/(1) = -j2 Step 2: Simplify the first term, (2+j3)/(5+j4), by multiplying the numerator and denominator by the conjugate of the denominator, which is 5-j4. (2+j3)/(5+j4) = (2+j3)/(5+j4) × (5-j4)/(5-j4) Numerator: (2+j3)(5-j4) = 2(5) + 2(-j4) + j3(5) + j3(-j4) = 10 - j8 + j15 - j^2 12 = 10 + j7 - (-1)12 = 10 + j7 + 12 = 22 + j7 Denominator: (5+j4)(5-j4) = 5^2 - (j4)^2 = 25 - j^2 16 = 25 - (-1)16 = 25 + 16 = 41 So the first term simplifies to (22+j7)/(41) = (22)/(41) + j(7)/(41). Step 3: Add the simplified first term and the simplified second term. ((22)/(41) + j(7)/(41)) + (-j2) = (22)/(41) + j(7)/(41) - j(82)/(41) = (22)/(41) + j((7-82)/(41)) = (22)/(41) - j(75)/(41) The expression in the form a+jb is 2241 - j7541. Problem 3: If z = (1)/(2+j3) + (1)/(1-j2), express z in the form a+jb. Step 1: Simplify the first term, (1)/(2+j3), by multiplying the numerator and denominator by its conjugate, 2-j3. (1)/(2+j3) = (1)/(2+j3) × (2-j3)/(2-j3) = (2-j3)/(2^2 - (j3)^2) = (2-j3)/(4 - j^2 9) = (2-j3)/(4 - (-1)9) = (2-j3)/(4+9) = (2-j3)/(13) = (2)/(13) - j(3)/(13) Step 2: Simplify the second term, (1)/(1-j2), by multiplying the numerator and denominator by its conjugate, 1+j2. (1)/(1-j2) = (1)/(1-j2) × (1+j2)/(1+j2) = (1+j2)/(1^2 - (j2)^2) = (1+j2)/(1 - j^2 4) = (1+j2)/(1 - (-1)4) = (1+j2)/(1+4) = (1+j2)/(5) = (1)/(5) + j(2)/(5) Step 3: Add the simplified first and second terms. z = ((2)/(13) - j(3)/(13)) + ((1)/(5) + j(2)/(5)) Combine the real parts and the imaginary parts: z = ((2)/(13) + (1)/(5)) + j(-(3)/(13) + (2)/(5)) Find a common denominator for the real parts (13 and 5, LCM is 65): (2)/(13) + (1)/(5) = (2 × 5)/(13 × 5) + (1 × 13)/(5 × 13) = (10)/(65) + (13)/(65) = (23)/(65) Find a common denominator for the imaginary parts (13 and 5, LCM is 65): -(3)/(13) + (2)/(5) = (-3 × 5)/(13 × 5) + (2 × 13)/(5 × 13) = (-15)/(65) + (26)/(65) = (11)/(65) So, z = (23)/(65) + j(11)/(65). The expression for z in the form a+jb is 2365 + j1165. That's 2 down. 3 left today — send the next one.