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(A) The reaction with $AgNO_3$ proceeds via the formation of a carbocation intermediate ($S_N1$ mechanism).The rate of reaction is directly proportion

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$(i) > (ii) > (iii)$

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(A) The reaction with $AgNO_3$ proceeds via the formation of a carbocation intermediate ($S_N1$ mechanism).The rate of reaction is directly proportional to the stability of the carbocation formed after the removal of the $Br^-$ ion.$(i)$ Forms an allylic carbocation,which is resonance-stabilized.(ii) Forms a vinylic carbocation,which is highly unstable due to the positive charge on an $sp^2$ hybridized carbon.(iii) Forms a secondary $(2^{\circ})$ alkyl carbocation,which is more stable than a vinylic carbocation but less stable than an allylic carbocation.Therefore,the stability order of the carbocations is: Allylic > Secondary alkyl > Vinylic.The rate of reaction follows the same order: $(i) > (iii) > (ii)$.

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