= Solution
If $X$ is <connected>, its <continuous> <surjective> image under $q$ is <connected> by (a)(ii). Conversely suppose $X/R$ is <connected> and $A\subseteq X$ is <clopen>. By (i) both $A$ and its complement are saturated. The quotient definition then makes $q(A)$ open, and its complement $q(X\setminus A)$ open too. Thus $q(A)$ is <clopen> in the <connected> quotient, so is empty or the whole quotient. Pulling back gives $A=\varnothing$ or $X$. By the <clopen> criterion $X$ is <connected>. Hence $\boxed{X\text{ connected}\iff X/R\text{ connected}}$.
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