$\left(\cos \frac{\pi}{2}+i \sin \frac{\pi}{2}\right) \times \left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right) \times \left(\cos \frac{\pi}{8}+i \sin \frac{\pi}{8}\right) \times \ldots \infty =$

Let $Z$ and $W$ be complex numbers such that $|Z| = |W|$,and $\text{arg } Z$ denotes the principal argument of $Z$.
Statement $1$: If $\text{arg } Z + \text{arg } W = \pi$,then $Z = -\overline{W}$.
Statement $2$: $|Z| = |W|$ implies $\text{arg } Z - \text{arg } \overline{W} = \pi$.

The modulus-amplitude form of $\frac{(1-i)^3(2-i)}{(2+i)(1+i)}$ is

Let $A = \{z : (\frac{z - \bar{z}}{2i})^2 \leqslant 2(\frac{z - \bar{z}}{2i})\}$ where $i = \sqrt{-1}$ and $B = \{z : |z| \leqslant \sqrt{5}\}$. The number of points with integral real and imaginary parts of $z$ lying in $A \cap B$ is -

The biquadratic equation,two of whose roots are $1+i$ and $1-\sqrt{2}$,is

If $z$ is a complex number such that $\left|z-\frac{4}{z}\right|=2$,then the greatest value of $|z|$ is