Let $\mathrm{z}=\mathrm{a}+ib, \mathrm{b}\ne 0$ be complex numbers satisfying ${\mathrm{z}}^2=\overline{\mathrm{z}}·2^{1-\left|z\right|}$. Then the least value of $n\in N$, such that $z^n={\left(z+1\right)}^n$, is equal to _____ .

Let $z_1=2+3i$ and $z_2=3+4i$. The set $\mathrm{S}=\left\{\mathrm{z}\in \mathrm{C}:{\left|\mathrm{z}-{\mathrm{z}}_1\right|}^2-{\left|\mathrm{z}-{\mathrm{z}}_2\right|}^2={\left|{\mathrm{z}}_1-{\mathrm{z}}_2\right|}^2\right\}$ represents a

For two non-zero complex number $z_1$ and $z_2$, if $Re\left(z_1{z}_2\right)=0$ and $Re\left(z_1+z_2\right)=0$, then which of the following are possible?

(A) $Im\left({\mathrm{z}}_1\right)>0$ and $Im\left({\mathrm{z}}_2\right)>0$
(B) $Im\left({\mathrm{z}}_1\right)<0$ and $Im\left({\mathrm{z}}_2\right)>0$
(C) $Im\left({\mathrm{z}}_1\right)>0$ and $Im\left({\mathrm{z}}_2\right)<0$
(D) $Im\left({\mathrm{z}}_1\right)<0$ and $Im\left({\mathrm{z}}_2\right)<0$

Choose the correct answer from the options given below: