A) \[-i\sqrt{3}\]
B) \[i\sqrt{3}\]
C) \[1-i\sqrt{3}\]
D) \[-1+i\sqrt{3}\]
Correct Answer: B
Solution :
\[4+5{{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{2008}}+3{{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{2009}}\] \[=4+5{{(\omega )}^{3\times 609+1}}+3{{(\omega )}^{3\times 609+2}}\] \[=4+5\omega +3{{\omega }^{2}}\] \[[\because \,\,{{\omega }^{3}}=1]\] \[=4+2\omega +3(\omega +{{\omega }^{2}})\] \[=4+2\left( \frac{-1+i\sqrt{3}}{2} \right)-3\] \[[\because \,\,1+\omega +{{\omega }^{2}}=0]\] \[=i\sqrt{3}\]
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