Find the number of non-zero integral solutions of the equation $|1-i|^{x}=2^{x}$.

If complex numbers $(x - 2y) + i(3x - y)$ and $(2x - y) + i(x - y + 6)$ are conjugates of each other,then $|x + iy|$ is $(x, y \in \mathbb{R})$.

If $\log_{\tan 30^{\circ}} \left( \frac{2|z|^2 + 2|z| - 3}{|z| + 1} \right) < -2$,then:

If $\frac{(p + i)^2}{2p - i} = \mu + i\lambda$,then $\mu^2 + \lambda^2$ is equal to

The complex number $z$ satisfies the condition $\left| z - \frac{25}{z} \right| = 24$. The maximum distance from the origin of coordinates to the point $z$ is:

If $\alpha_1, \alpha_2, \alpha_3$ respectively denote the moduli of the complex numbers $-i, \frac{1}{3}(1+i)$ and $-1+i$,then their increasing order is