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Question
\[\frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\] equals
Options
i
-1
\[-\]i
4
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Solution
\[-\]i
\[\text { Let z } = \frac{1 + 2i + 3 i^2}{1 - 2i + 3 i^2}\]
\[ \Rightarrow z = \frac{1 + 2i - 3}{1 - 2i - 3}\]
\[ \Rightarrow z=\frac{- 2 + 2i}{- 2 - 2i}\times$\frac{- 2 + 2i}{- 2 + 2i}\]
\[ \Rightarrow z=\frac{\left( - 2 + 2i \right)^2}{\left( - 2 \right)^2 - \left( 2i \right)^2}\]
\[ \Rightarrow z=\frac{4 + 4 i^2 - 8i}{4 + 4}\]
\[ \Rightarrow z =\frac{4 - 4 - 8i}{8}\]
\[ \Rightarrow z=\frac{- 8i}{8}\]
\[ \Rightarrow z =-i\]
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