Two concentric shells have radii R and 2R, charges ${\mathrm{q}}_{\mathrm{A}}$ and ${\mathrm{q}}_{\mathrm{B}}$ and potentials 2V and $\left(\frac{3}{2}\right)\mathrm{V},$ respectively. Now, shell B is earthed and let charges on them become ${\mathrm{q}\text{'}}_{\mathrm{A}} \mathrm{and} {\mathrm{q}\text{'}}_{\mathrm{B}}$. Then 

Before earthing:

${\mathrm{V}}_{\mathrm{A}}=2\mathrm{V}=\mathrm{k}\left[\frac{{\mathrm{q}}_{\mathrm{A}}}{\mathrm{R}}+\frac{{\mathrm{q}}_{\mathrm{B}}}{2\mathrm{R}}\right] .... \left(1\right)$

${\mathrm{V}}_{\mathrm{B}}=\frac{3}{2}\mathrm{V}=\mathrm{k}\left[\frac{{\mathrm{q}}_{\mathrm{A}}+{\mathrm{q}}_{\mathrm{B}}}{2\mathrm{R}}\right] .... \left(2\right)$

Dividing eq(1) by eq(2), we get

$\frac{4}{3}=\frac{2{\mathrm{q}}_{\mathrm{A}}+{\mathrm{q}}_{\mathrm{B}}}{{\mathrm{q}}_{\mathrm{A}}+{\mathrm{q}}_{\mathrm{B}}}\Rightarrow \frac{{\mathrm{q}}_{\mathrm{A}}}{{\mathrm{q}}_{\mathrm{B}}}=\frac{1}{2}$

After earthing,

${\mathrm{V}}_{\mathrm{B}}=0=\mathrm{k}\left[\frac{{\mathrm{q}}_{\mathrm{A}}^{\text{'}}+{\mathrm{q}}_{\mathrm{B}}^{\text{'}}}{2\mathrm{R}}\right]\Rightarrow {\mathrm{q}}_{\mathrm{A}}^{\text{'}}=-{\mathrm{q}}_{\mathrm{B}}^{\text{'}}$

$\Rightarrow \left|\frac{{\mathrm{q}}_{\mathrm{A}}^{\mathrm{\prime }}}{{\mathrm{q}}_{\mathrm{B}}^{\text{'}}}\right|=1$

Potential difference between A and B:

${\mathrm{V}}_{\mathrm{A}}^{\mathrm{\prime }}-{\mathrm{V}}_{\mathrm{B}}^{\mathrm{\prime }}={\mathrm{V}}_{\mathrm{A}}-{\mathrm{V}}_{\mathrm{B}}=2\mathrm{V}-\frac{3}{2}\mathrm{V}=\frac{1}{2}\mathrm{V}$