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  4. 4+5( (-1+i√3/2) )2008+3( (-1+i√3/2) )2009 is equal to

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Q. $ 4+5{{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{2008}}+3{{\left( \frac{-1+i\sqrt{3}}{2} \right)}^{2009}} $ is equal to

J & K CETJ & K CET 2009Complex Numbers and Quadratic Equations

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} $