Determine the type of signal $x(t)$ given by his coefficients [ Fourier Series ]

Question

I want to determine if the signal $x(t)$ is

1. even
2. real
3. $\frac{dx(t)}{dt}$ is even
   

By the following coefficients

$$C_k =\begin{cases} 2 & k=0 \\ j(\frac{1}{2})^{|k|} & else \end{cases}$$