For this question, you must not use a calculator. The complex number $u$ is given by $u = \frac{1 + 2i}{1 - 3i}$.
(i)[3]
Express $u$ as $x + iy$, where $x$ and $y$ are real.
(ii)[2]
On a sketch of an Argand diagram, indicate the points $A$, $B$ and $C$ for the complex numbers $u$, $1 + 2i$ and $1 - 3i$ respectively.
(iii)[3]
By using the arguments of $1 + 2i$ and $1 - 3i$, show that $\tan^{-1} 2 + \tan^{-1} 3 = \frac{3}{4}\pi$.
Worked solution & mark scheme
This 8-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Multiply the numerator and denominator by $1 + 3i$, or an equivalent factor” …