[tex]z_1=\sqrt{3}\left(cos\:\frac{\pi }{4}+i\:sin\:\frac{\pi }{4}\right)[/tex]
[tex]z_2=\sqrt{6}\left(cos\:\frac{3\pi \:}{4}+i\:sin\:\frac{3\pi \:}{4}\right)[/tex]
[tex]\cos \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2}[/tex]
[tex]\sin \left(\frac{\pi }{4}\right)=\frac{\sqrt{2}}{2}[/tex]
[tex]\cos \left(\frac{3\pi }{4}\right)=-\frac{\sqrt{2}}{2}[/tex]
[tex]\sin \left(\frac{3\pi }{4}\right)=\frac{\sqrt{2}}{2}\\[/tex]
[tex]Z_1*Z_2=\sqrt{3}\left(cos\:\frac{\pi }{4}+i\:sin\:\frac{\pi }{4}\right)\cdot \sqrt{6}\left(cos\:\frac{3\pi \:}{4}+i\:sin\:\frac{3\pi \:}{4}\right)[/tex]
[tex]=\sqrt{3}\left(\frac{\sqrt{2}}{2}+\:i\frac{\sqrt{2}}{2}\right)\cdot \sqrt{6}\left(\frac{-\sqrt{2}}{2}+\:i\frac{\sqrt{2}}{2}\right)[/tex]
On simplifying, we get
[tex]Z_1* Z_2 =-3\sqrt{2}[/tex]
Therefore, correct option is 1st option.