A) 0
B) 1
C) i
D) \[-i\]
Correct Answer: C
Solution :
Consider \[{{(-\sqrt{-1})}^{4n+3}}\,+{{({{i}^{41}}-{{i}^{-257}})}^{9}}\] \[=\,{{(-i)}^{4n+3}}+{{\left[ {{({{i}^{4}})}^{10}}.\,{{i}^{1}}+{{({{i}^{3}})}^{-85}}.\,{{i}^{-\,2}} \right]}^{9}}\] \[=\,\,{{(-i)}^{4n+3}}+{{\left[ i+\frac{1}{{{({{i}^{3}})}^{85}}}.\frac{1}{{{i}^{2}}} \right]}^{9}}\] \[=\,\,{{(-i)}^{4n+3}}+{{\left( i+\frac{1}{i} \right)}^{9}}\] \[=\,\,-{{(-1)}^{4n+3}}\,{{(i)}^{4n}}\,{{(i)}^{3}}\,+{{(i\,-\,\,i)}^{9}}=\,\,-(1)\,(-i)+0\,\,=\,\,i\]
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