Moderate -0.5 This is a straightforward application of the definition of argument using tan(arg z) = Im(z)/Re(z), followed by direct calculation of modulus and conjugate. While it's a Further Maths topic, the question requires only routine recall and basic algebraic manipulation with no problem-solving insight needed, making it easier than average.
1 The complex number \(3 + a \mathrm { i }\), where \(a\) is real, is denoted by \(z\). Given that \(\arg z = \frac { 1 } { 6 } \pi\), find the value of \(a\) and hence find \(| z |\) and \(z ^ { * } - 3\).
1 The complex number $3 + a \mathrm { i }$, where $a$ is real, is denoted by $z$. Given that $\arg z = \frac { 1 } { 6 } \pi$, find the value of $a$ and hence find $| z |$ and $z ^ { * } - 3$.
\hfill \mbox{\textit{OCR FP1 2013 Q1 [6]}}