If $$e^{i \theta}=\operatorname{cis} \theta$$, then $$\sum_\limits{n=0}^{\infty} \frac{\cos (n \theta)}{2^n}=$$

$$i z^3+z^2-z+i=0 \Rightarrow|z|=$$

If $$\frac{x-1}{3+i}+\frac{y-1}{3-i}=i$$, then the true statement among the following is

The number of integer solutions of the equation $$|1-i|^x=2^x$$ is