Which of the following compounds will form the precipitate with aq .   AgNO 3 solution most readily?

It is an example of ${\mathrm{S}}_{\mathrm{N}}1$ reaction.

Rate of ${\mathrm{S}}_{\mathrm{N}}1 \mathrm{reaction} \mathrm{\alpha } \mathrm{stability} \mathrm{of} \mathrm{carbocation}$

In option 1 carbocation formed after removing halogen is destabilized by $-\mathrm{I}$ effect.

In option 2 carbocation formed after removing halogen is stabilized by +M effect since lone pair is present at oxygen atom.

In option 3 carbocation formed after removing halogen is stabilized by +M effect of lone pair present on nitrogen atom but lone pair is also in conjugation with benzene ring therefore +M effect is less.

In option 4 carbocation formed after removing halogen is stabilized by +M effect of lone pair present on nitrogen atom and +M effect of nitrogen atom is more than that of oxygen atom. Hence, option 4 is most stable.