Use complex numbers in the form 𝑎 + 𝑏𝑖 where 𝑎 and 𝑏 are the real and imaginary parts (components) Re(𝑧) and Im(𝑧) of a complex number 𝑧.

Let \(z_1 = 3 - 5i\) and \(z_2 = -2 + 7i\). (a) Determine \(z_1 + 2z_2\), expressing your answer in the form \(a + bi\). (1 mark) (b) Find \(\text{Re}(z_1 z_2)\). (1 mark) (c) Calculate \(\text{Im}\left(\frac{z_1}{z_2}\right)\), giving your answer as a simplified fraction. (1 mark)

A complex number \(z\) is plotted on an Argand diagram with real axis from \(−4\) to \(4\) and imaginary axis from \(−3\) to \(3\). The point is located 2 units to the right of the origin and 1 unit down from the real axis. Determine the imaginary part of \(z\).

A research team is analysing oscillating electrical signals in a circuit. The voltage at time \(t\) seconds is modelled by the complex number \(V = (3 + 2i)(4 - i) + \frac{6 + 8i}{1 - i}\), where the real part represents the in-phase component (in volts) and the imaginary part represents the quadrature component (in volts). (a) Determine the complex number \(V\) in the form \(a + bi\). (2 marks) (b) State the value of the quadrature component of the voltage. (1 mark)