The focal length of a planoconvex glass lens is $20 \,cm$ $(\mu_g=1.5)$. The plane face of it is silvered. An illuminating object is placed at a distance of $60 \,cm$ from the lens on its axis along the convex side. Then the distance (in $cm$) of the image is .........

$A$ converging mirror is placed on the right-hand side of a converging lens as shown in the figure. The focal length of the lens and the mirror are $15 \ cm$ and $20 \ cm$ respectively. The separation between the lens and the mirror is $40 \ cm$ and their principal axes coincide. $A$ point source is placed on the principal axis at a distance $d$ to the left of the lens. If the final beam comes out parallel to the principal axis, then the value of $d$ is: (in $cm$)

The figure shows an equiconvex lens (of refractive index $1.50$) in contact with a liquid layer on top of a plane mirror. $A$ small needle with its tip on the principal axis is moved along the axis until its inverted image is found at the position of the needle. The distance of the needle from the lens is measured to be $45.0 \; cm$. The liquid is removed and the experiment is repeated. The new distance is measured to be $30.0 \; cm$. What is the refractive index of the liquid?

When the plane surface of a plano-convex lens is silvered,it behaves as a concave mirror of focal length $60 \ cm$. However,when the convex surface is silvered,it behaves as a concave mirror of focal length $20 \ cm$. What is the refractive index of the lens?

When a thin convex lens is placed in contact with a concave mirror of the same focal length,the resultant focal length of the system is ......

$A$ convex lens forms a real image of a point object at a distance of $50 \, cm$ from the convex lens. $A$ concave lens is placed $10 \, cm$ behind the convex lens on the image side. On placing a plane mirror on the image side and facing the concave lens,it is observed that the final image now coincides with the object itself. The focal length of the concave lens is .............. $cm$.