Let $$Z_1, Z_2$$ and $$Z_3$$ be three non zero complex numbers such that $$a=\left|Z_1\right|, b=\left|Z_2\right|$$ and $$c=\left|Z_3\right|$$, if the determinant $$\left|\begin{array}{lll}a & b & c \\ b & c & a \\ c & a & b\end{array}\right|=0$$, then

If $$\left|z_1+z_2\right|^2=\left|z_1\right|^2+\left|z_2\right|^2$$, where $$z_1$$ and $$z_2$$ are two complex numbers, then

A real value of $$x$$ will satisfy the equation, $$\left(\frac{3-4 i x}{3+4 i x}\right)=\alpha-i \beta,(\alpha, \beta$$ are real $$)$$, if

What is the value of $$(1-i \sqrt{3})^9$$ is equal to