A) \[|z|\,=1,\,\,\,\,arg\,z=\frac{\pi }{4}\]
B) \[|z|\,=1,arg\,z=\frac{\pi }{6}\]
C) \[|z|\,=\frac{\sqrt{3}}{2},\,arg\,z=\frac{5\pi }{24}\]
D) \[|z|\,=\frac{\sqrt{3}}{2},\,\,arg\,z={{\tan }^{-1}}\frac{1}{\sqrt{2}}\]
Correct Answer: B
Solution :
\[z=\cos \frac{\pi }{6}+i\sin \frac{\pi }{6}=\frac{\sqrt{3}}{2}+\frac{i}{2}\] \[\therefore \,\,|z|\,=\sqrt{\frac{3}{4}+\frac{1}{4}}=1\] and \[arg\,(z)={{\tan }^{-1}}\,\left( \frac{y}{x} \right)={{\tan }^{-1}}\left( \frac{1/2}{\sqrt{3}/2} \right)={{\tan }^{-1}}\left( \frac{1}{\sqrt{3}} \right)\] \[\Rightarrow \,\,arg(z)\,={{\tan }^{-1}}\left( \tan \frac{\pi }{6} \right)=\frac{\pi }{6}\].
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