A planoconvex lens when silvered on the plane side behaves like a concave mirror of focal length 60 cm. However. when silvered on the convex side it behaves like a concave mirror of focal length 20 cm. Then the refractive index of the lens is

$\begin{array}{l}\frac{1}{\mathrm{F}}=\frac{1}{{\mathrm{f}}_1}+\frac{1}{{\mathrm{f}}_1}+\frac{1}{{\mathrm{f}}_{\mathrm{m}}}\\ \frac{1}{\mathrm{F}}=\frac{2}{{\mathrm{f}}_1}+\frac{1}{{\mathrm{f}}_{\mathrm{m}}}\\ \mathrm{or}\frac{1}{\mathrm{F}}=2\left(\mathrm{\mu }-1\right)\left(\frac{1}{\mathrm{R}}\right)+\frac{1}{\mathrm{\infty }}\\ \mathrm{or}\mathrm{F}=\frac{\mathrm{R}}{2(\mathrm{\mu }-1)}\end{array}$

$\begin{array}{l}\mathrm{Now},-60=\frac{-\mathrm{R}}{2\left(\mathrm{\mu }-1\right)}\\ \mathrm{or}120\left(\mathrm{\mu }-1\right)=\mathrm{R}....\left(1\right)\\ \mathrm{Again},\frac{1}{\mathrm{F}}=\frac{1}{{\mathrm{f}}_1}+\frac{1}{{\mathrm{f}}_1}+\frac{1}{fm}\\ \mathrm{or}\frac{1}{\mathrm{F}}=\frac{2}{{\mathrm{f}}_1}+\frac{1}{\mathrm{R}/2}\end{array}$

$\begin{array}{l}\mathrm{or}\frac{1}{\mathrm{F}}=2\left(\mathrm{\mu }-1\right)\left(\frac{1}{\mathrm{R}}\right)+\frac{2}{\mathrm{R}}\\ \mathrm{or}\frac{1}{\mathrm{F}}=\frac{2}{\mathrm{F}}\left(\mathrm{\mu }-1+1\right)\\ \mathrm{or}\mathrm{F}=\frac{\mathrm{R}}{2\mathrm{\mu }}\\ \mathrm{Now},-20=\frac{-\mathrm{R}}{2\mathrm{\mu }}\mathrm{or}40\mathrm{\mu }=\mathrm{R}...\left(2\right)\\ \mathrm{Dividing}\left(1\right)\mathrm{by}\left(2\right),\mathrm{we}\mathrm{get}\\ \frac{120\left(\mathrm{\mu }-1\right)}{40\mathrm{\mu }}=\frac{\mathrm{R}}{\mathrm{R}}=1\\ \mathrm{or}120\left(\mathrm{\mu }-1\right)=40\mathrm{\mu }\\ \mathrm{or}120\mathrm{\mu }-40\mathrm{\mu }=120\\ \mathrm{or}80\mathrm{\mu }=120\\ \mathrm{or}\mathrm{\mu }=\frac{120}{80}=\frac{3}{2}=1.5\end{array}$