$20pt$ Determine whether the signal shown below satisfies the following conditions

$(1)Re\{x(j)\}=0$

$(2)In\{x(j)\}=0$

$(3)$ There exists a real $$ such that $e^{j}x(j)$ is real

$(4){\displaystyle \underset{-}{\overset{}{}}}x(j)d=0$

$(5){\displaystyle \underset{-}{\overset{}{}}}x(j)d=0$

$(6)x(j)$ is periodic