A) \[R\left( z \right)=-3\]
B) \[R\left( z \right)<0\text{ }and\text{ }I\left( z \right)>0\]
C) \[I\left( z \right)=0\]
D) \[R\left( z \right)>0\text{ }and\text{ }I\left( z \right)>0\]
Correct Answer: C
Solution :
\[z={{\left( \frac{\sqrt{3}}{2}+\frac{i}{2} \right)}^{5}}+{{\left( \frac{\sqrt{3}}{2}-\frac{i}{2} \right)}^{5}}\] \[z={{\left( {{e}^{i\frac{\pi }{6}}} \right)}^{5}}+{{\left( {{e}^{-i\frac{\pi }{6}}} \right)}^{5}}\] \[z\,\,=\,{{e}^{i\frac{5\pi }{6}}}+{{e}^{-i\frac{5\pi }{6}}}\] \[z\,\,=\,\,2\cos \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{5\pi }{6}\] \[z\,\,=\,\,2\left( -\frac{\sqrt{3}}{2} \right)\,=-\sqrt{3}\,\,\,\,\,\,\] It means \[Im\left( z \right)=0\]You need to login to perform this action.
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