A) 0
B) 1
C) 84
D) 206
Correct Answer: A
Solution :
(a): \[{{\left( a+\frac{1}{a} \right)}^{2}}=3\,\,\,\Rightarrow a+\frac{1}{a}=\sqrt{3}\] On cubing both sides, \[{{a}^{3}}+\frac{1}{{{a}^{3}}}+3\left( a+\frac{1}{a} \right)=3\sqrt{3}\] \[\Rightarrow {{a}^{3}}+\frac{1}{{{a}^{3}}}=3\sqrt{3}-3\sqrt{3}=0\,\,\,\,\,\,\Rightarrow {{a}^{6}}+1=0\] \[\therefore {{a}^{200}}+{{a}^{200}}+{{a}^{90}}+{{a}^{84}}+{{a}^{18}}+{{a}^{12}}+{{a}^{6}}+1\] \[={{a}^{200}}\left( {{a}^{6}}+1 \right)+{{a}^{84}}\left( {{a}^{6}}+1 \right)+{{a}^{12}}\left( {{a}^{6}}+1 \right)+\left( {{a}^{6}}+1 \right)\]\[=0\]
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