Given z = (5)/(2-i) - (3+2i), find z2 and its modulus and argument. Then, solve the equation (z)/(1+2i) + (z-1)/(5) = (1)/(1-2i) for z.

Here's the solution for question 5: i) Given z = (5)/(2-i) - (3+2i). Step 1: Express the first term in the form a+bi. (5)/(2-i) = (5)/(2-i) × (2+i)/(2+i) = (5(2+i))/(2^2 - i^2) = (10+5i)/(4 - (-1)) = (10+5i)/(5) = 2+i Step 2: Substitute this back into the expression for z and simplify. z = (2+i) - (3+2i) = 2+i-3-2i = (2-3) + (1-2)i = -1-i So, z in the form x+iy is -1-i. Step 3: Calculate z^2. z^2 = (-1-i)^2 = (-(1+i))^2 = (1+i)^2 = 1^2 + 2(1)(i) + i^2 = 1+2i-1 = 2i Step 4: Find |z^2|. For z^2 = 0+2i, the modulus is: |z^2| = sqrt(0^2 + 2^2) = sqrt(4) = 2 So, |z^2| = 2. Step 5: Find z^2. For z^2 = 2i, the complex number lies on the positive imaginary axis. z^2 = ()/(2) So, z^2 = ()/(2). ii) Given that (z)/(1+2i) + (z-1)/(5) = (1)/(1-2i). Step 1: Simplify the terms involving complex denominators. (z)/(1+2i) = (z(1-2i))/((1+2i)(1-2i)) = (z(1-2i))/(1^2 - (2i)^2) = (z(1-2i))/(1+4) = (z(1-2i))/(5) (1)/(1-2i) = (1(1+2i))/((1-2i)(1+2i)) = (1+2i)/(1^2 - (2i)^2) = (1+2i)/(1+4) = (1+2i)/(5) Step 2: Substitute these simplified terms back into the equation. (z(1-2i))/(5) + (z-1)/(5) = (1+2i)/(5) Step 3: Multiply the entire equation by 5 to clear the denominators. z(1-2i) + (z-1) = 1+2i z - 2iz + z - 1 = 1+2i 2z - 2iz - 1 = 1+2i z(2-2i) = 1+2i+1 z(2-2i) = 2+2i Step 4: Solve for z. z = (2+2i)/(2-2i) = (2(1+i))/(2(1-i)) = (1+i)/(1-i) Multiply by the conjugate of the denominator: z = (1+i)/(1-i) × (1+i)/(1+i) = ((1+i)^2)/(1^2 - i^2) = (1+2i+i^2)/(1 - (-1)) = (1+2i-1)/(2) = (2i)/(2) = i So, z=i. Step 5: Find real numbers p and q such that (p+iq)z = 3+4i. Substitute z=i: (p+iq)i = 3+4i pi + i^2q = 3+4i pi - q = 3+4i Rearrange to group real and imaginary parts: -q + pi = 3+4i Equating the real parts: -q = 3 q = -3 Equating the imaginary parts: p = 4 So, p = 4 and q = -3. iii) Find the general solution of the equation 2x + 1 = 2x. Step 1: Use the double angle identity 2x = 2^2 x - 1. (2^2 x - 1) + 1 = 2x 2^2 x = 2x Step 2: Use the double angle identity 2x = 2 x x. 2^2 x = 2 x x Step 3: Rearrange the equation and factor. 2^2 x - 2 x x = 0 2 x ( x - x) = 0 Step 4: Solve for the two possible cases. Case 1: 2 x = 0 x = 0 The general solution for x = 0 is x = ()/(2) + n, where n Z. Case 2: x - x = 0 x = x Divide by x (assuming $ x ≠