Standard +0.3 This is a straightforward complex number problem requiring algebraic manipulation to find k, then converting to modulus-argument form. The condition that z + 12/z is real leads directly to equating imaginary parts to zero, giving a simple equation. While it involves multiple steps (finding k, then |z| and arg(z)), each step uses standard techniques with no novel insight required, making it slightly easier than average.
The complex number \(z\) is given by \(z = k + 3i\), where \(k\) is a negative real number.
Given that \(z + \frac{12}{z}\) is real, find \(k\) and express \(z\) in exact modulus-argument form. [6]
The complex number $z$ is given by $z = k + 3i$, where $k$ is a negative real number.
Given that $z + \frac{12}{z}$ is real, find $k$ and express $z$ in exact modulus-argument form. [6]
\hfill \mbox{\textit{SPS SPS FM Pure 2023 Q10 [6]}}