A) 20
B) 9
C) \[\frac{5}{4}\]
D) \[\frac{4}{5}\]
Correct Answer: D
Solution :
\[\left| \frac{1+i\sqrt{3}}{{{\left( 1+\frac{1}{i+1} \right)}^{2}}} \right|=\left| \frac{(1+i\sqrt{3})\,{{(i+1)}^{2}}}{{{(i+2)}^{2}}} \right|\,\,(\because \,\,{{i}^{2}}=-1)\] \[=\left| \frac{(1+i\sqrt{3})\,({{i}^{2}}+1+2i)}{({{i}^{2}}+4+4i)} \right|\] \[=\left| \frac{(1+i\sqrt{3})2i}{(4i+3)} \right|\] \[=\left| \frac{-2\sqrt{3}+2i}{3+4i} \right|\] \[=2\left| \frac{(-\sqrt{3}+i)\,(3-4i)}{(3+4i)\,(3-4i)} \right|\] \[=2\left| \frac{-3\sqrt{3}+3i+4\sqrt{3}i+4}{25} \right|\] \[=\frac{2}{25}|(4-3\sqrt{3})+(3+4\sqrt{3})i|\] \[=\frac{2}{25}\sqrt{{{(4-3\sqrt{3})}^{2}}+{{(3+4\sqrt{3})}^{2}}}\] \[=\frac{2}{25}\sqrt{16+27-24\sqrt{3}+9+48+24\sqrt{3}}\] \[=\frac{2}{25}\sqrt{100}=\frac{2\times 10}{25}=\frac{4}{5}\]
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