Question
If $$z = {\left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right)^5} + {\left( {\frac{{\sqrt 3 }}{2} - \frac{i}{2}} \right)^5},\,{\text{then}}$$
A.
$${\text{Re}}\left( z \right) = 0$$
B.
$${\text{Im}}\left( z \right) = 0$$
C.
$${\text{Re}}\left( z \right) > 0,{\text{Im}}\left( z \right) > 0$$
D.
$${\text{Re}}\left( z \right) > 0,{\text{Im}}\left( z \right) < 0$$
Answer :
$${\text{Im}}\left( z \right) = 0$$
Solution :
$$\eqalign{
& \left( {\frac{{\sqrt 3 }}{2} + \frac{i}{2}} \right) = - i\left( {\frac{{ - 1}}{2} + \frac{{i\sqrt 3 }}{2}} \right) = i\omega \cr
& \frac{{\sqrt 3 }}{2} - \frac{i}{2} = i\left( {\frac{{ - 1}}{2} - \frac{{i\sqrt 3 }}{2}} \right) = i{\omega ^2} \cr
& \therefore \,\,z = {\left( { - i\omega } \right)^5} + {\left( {i{\omega ^2}} \right)^5} = - i{\omega ^2} + i\omega \cr
& = i\left( {\omega - {\omega ^2}} \right) = i\left( {i\sqrt 3 } \right) = - \sqrt 3 \cr
& \Rightarrow \,\,{\text{Re}}\left( z \right) < 0\,\,{\text{and Im}}\left( z \right) = 0 \cr
& \therefore \,\,\,\left( {\text{B}} \right)\,\,{\text{is the correct choice}}{\text{.}} \cr} $$