$$ \text { If }(1-4 i)^3=a+i b \text { then the value of } \mathrm{a} \text { and } \mathrm{b} \text { is } $$

$$ \text { The value of } \frac{i^{1004}+i^{1006}+i^{1008}+i^{1010}+i^{1012}}{i^{510}+i^{508}+i^{506}+i^{504}+i^{502}} \text { is } $$

$$ \text { The modulus of the following complex number } \frac{1+i}{1-i}-\frac{1-i}{1+i} \text { is } $$

If $$z=\sqrt{3}+i$$, then the argument of $$z^2 e^{z-i}$$ is equal to