A) \[|{{z}_{1}}{{z}_{2}}{{z}_{3}}.....{{z}_{n}}|\]
B) \[|{{z}_{1}}|+|{{z}_{2}}|+....+|{{z}_{n}}|\]
C) \[\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+....+\frac{1}{{{z}_{n}}} \right|\]
D) \[n\]
E) \[\sqrt{n}\]
Correct Answer: D
Solution :
We have, \[|{{z}_{1}}|=|{{z}_{2}}|=....=|{{z}_{n}}|=1\] \[\Rightarrow \] \[{{z}_{1}}\overline{{{z}_{1}}}={{z}_{2}}\overline{{{z}_{2}}}=.....{{z}_{n}}\overline{{{z}_{n}}}=1\] \[\Rightarrow \] \[\overline{{{z}_{1}}}=\frac{1}{{{z}_{1}}},\overline{{{z}_{2}}}=\frac{1}{{{z}_{2}}},....\overline{{{z}_{n}}}=\frac{1}{{{z}_{n}}}\] Now, \[|{{z}_{1}}+{{z}_{2}}+....+{{z}_{n}}|\] \[=|\overline{{{z}_{1}}+{{z}_{2}}+....+{{z}_{n}}}|\] \[=|\overline{{{z}_{1}}}+\overline{{{z}_{2}}}+....+\overline{{{z}_{n}}}|=\left| \frac{1}{{{z}_{1}}}+\frac{1}{{{z}_{2}}}+....+\frac{1}{{{z}_{n}}} \right|\]
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