Normalize a symmetric law and average an absolute distance

The discrete random variable $X$ has probability function

$P(X=x)=\begin{cases}k(2-x),&x=0,1,2,\\k(x-2),&x=3,4,\\0,&\text{otherwise},\end{cases}$

where $k>0$. (i) Show that $k=1/6$ and state $E(X)$. (ii) Find $E(|X-2|)$. (iii) Find $\operatorname{Var}(X)$.