As, a filter paper soaked in salt $\text{X}$turns brown when exposed to $\text{H}\text{N}{\text{O}}_3$ vapour, then salt $\text{X}$ must be a strong reducing agent which will reduce $\text{H}\text{N}{\text{O}}_3$ to $\text{N}{\text{O}}_2$ (brown gas).

Among the given salt, $\text{K}\text{I}$ is the strongest reducing agent. Thus, salt $\text{X}$ is $\text{K}\text{I}$.

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$+2{\mathrm{H}}_2\mathrm{O}$