Challenging +1.2 This question requires substituting w = x + iy and w* = x - iy into the equation, then separating real and imaginary parts to form simultaneous equations. While it involves multiple algebraic steps and careful manipulation of complex conjugates, it follows a standard technique taught in P3. The constraint Re(w) ≤ 0 adds minimal difficulty as it simply filters solutions. More challenging than routine complex arithmetic but less demanding than problems requiring geometric insight or proof.
6 Find the complex numbers \(w\) which satisfy the equation \(w ^ { 2 } + 2 \mathrm { i } w ^ { * } = 1\) and are such that \(\operatorname { Re } w \leqslant 0\). Give your answers in the form \(x + \mathrm { i } y\), where \(x\) and \(y\) are real.
6 Find the complex numbers $w$ which satisfy the equation $w ^ { 2 } + 2 \mathrm { i } w ^ { * } = 1$ and are such that $\operatorname { Re } w \leqslant 0$. Give your answers in the form $x + \mathrm { i } y$, where $x$ and $y$ are real.\\
\hfill \mbox{\textit{CAIE P3 2022 Q6 [6]}}