A) \[1-i\sqrt{3}\]
B) \[-1+i\sqrt{3}\]
C) \[i\sqrt{3}\]
D) \[-i\sqrt{3}\]
Correct Answer: C
Solution :
\[4+5{{\left( -\frac{1}{2}+\frac{i\sqrt{3}}{2} \right)}^{334}}+3{{\left( -\frac{1}{2}+\frac{i\sqrt{3}}{2} \right)}^{365}}\] \[=4+5{{(\omega )}^{334}}+3{{(\omega )}^{365}}\] \[=4+5{{(\omega )}^{333}}\omega +3{{(\omega )}^{363}}{{\omega }^{2}}\] \[=4+5\omega +3{{\omega }^{2}}\] \[=4+3(\omega +{{\omega }^{2}})+2\omega \] \[=4+3(-1)+2\left( \frac{-1+i\sqrt{3}}{2} \right)\] \[=1-1+i\sqrt{3}\] \[=i\sqrt{3}\]
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