MCQ
If $z=\frac{1}{(1-i)(2+3 i)}$, then $|z|=$
- A1
- B$1 / \sqrt{26}$
- C$4 / \sqrt{26}$
- D$5/ \sqrt{26}$
✓
Answer
(b) $1 / \sqrt{26}$
Explanation: $1 / \sqrt{26}$
Let $z =\frac{1}{(1-i)(2+3 i)}$
$\begin{array}{l}\Rightarrow z=\frac{1}{2+i-3 i^2} \\ \Rightarrow z=\frac{1}{2+i+3} \\ \Rightarrow z=\frac{1}{5+i} \times \frac{5-i}{5-i} \\ \Rightarrow z=\frac{5-i}{25-i^2} \\ \Rightarrow z=\frac{5-i}{25+1} \\ \Rightarrow z=\frac{5-i}{26} \\ \Rightarrow z=\frac{5}{26}-\frac{i}{26} \\ \Rightarrow|z|=\sqrt{\frac{25}{676}+\frac{1}{676}} \\ \Rightarrow z=\frac{1}{\sqrt{26}}\end{array}$
Explanation: $1 / \sqrt{26}$
Let $z =\frac{1}{(1-i)(2+3 i)}$
$\begin{array}{l}\Rightarrow z=\frac{1}{2+i-3 i^2} \\ \Rightarrow z=\frac{1}{2+i+3} \\ \Rightarrow z=\frac{1}{5+i} \times \frac{5-i}{5-i} \\ \Rightarrow z=\frac{5-i}{25-i^2} \\ \Rightarrow z=\frac{5-i}{25+1} \\ \Rightarrow z=\frac{5-i}{26} \\ \Rightarrow z=\frac{5}{26}-\frac{i}{26} \\ \Rightarrow|z|=\sqrt{\frac{25}{676}+\frac{1}{676}} \\ \Rightarrow z=\frac{1}{\sqrt{26}}\end{array}$
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