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Question
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
Options
1
\[1/\sqrt{26}\]
\[5/\sqrt{26}\]
none of these
Solution
\[1/\sqrt{26}\]
\[\text { Let }z = \frac{1}{\left( 1 - i \right)\left( 2 + 3i \right)}\]
\[ \Rightarrow z = \frac{1}{2 + i - 3 i^2} \]
\[ \Rightarrow z = \frac{1}{2 + i + 3}\]
\[\Rightarrow z=\frac{1}{5 + i}\times\frac{5 - i}{5 - i}\]
\[\Rightarrow z=\frac{5 - i}{25 - i^2}\]
\[ \Rightarrow z=\frac{5 - i}{25 + 1}\]
\[\Rightarrow z=\frac{5 - i}{26}\]
\[\Rightarrow z = \frac{5}{26} - \frac{i}{26}\]
\[\Rightarrow \left| z \right|=\sqrt{\frac{25}{676} + \frac{1}{676}}\]
\[\Rightarrow z = \frac{1}{\sqrt{26}}\]
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