Question

\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals

Options

• i

• -1

• \[-\]i

• 4

Answer 1

\[-\]i

\[\text { Let z } = \frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\]

\[ \Rightarrow z = \frac{1 + 2i - 3}{1 - 2i - 3}\]

\[ \Rightarrow z=\frac{- 2 + 2i}{- 2 - 2i}\times$\frac{- 2 + 2i}{- 2 + 2i}\]

\[ \Rightarrow z=\frac{\left( - 2 + 2i \right)^2}{\left( - 2 \right)^2 - \left( 2i \right)^2}\]

\[ \Rightarrow z=\frac{4 + 4 i^2 - 8i}{4 + 4}\]

\[ \Rightarrow z =\frac{4 - 4 - 8i}{8}\]

\[ \Rightarrow z=\frac{- 8i}{8}\]

\[ \Rightarrow z =-i\]