$A$ plano-convex lens $(f=20 \,cm)$ is silvered at its plane surface. What will be its new focal length (in $\,cm$)?

An equiconvex glass lens $(a)$ has a focal length $f$ and power $P$. It is cut into two symmetrical halves $(b)$ by a plane containing the principal axis. The two pieces are recombined as shown in figure $(c)$. The power of the new combination is

$A$ concave mirror of focal length $f_{1}$ is placed at a distance $d$ from a convex lens of focal length $f_{2}$. $A$ parallel beam of light coming from infinity parallel to the principal axis falls on the convex lens and then after refraction falls on the concave mirror. If the light is to retrace its path,the distance $d$ should be:

The final image of the object forms on the object itself. If the focal length of the lens is $30 \ cm$,then the radius of curvature of the mirror is .... $cm$.

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 ......

Two identical thin plano-convex glass lenses (refractive index $1.5$) each having a radius of curvature of $20\, cm$ are placed with their convex surfaces in contact at the centre. The intervening space is filled with oil of refractive index $1.7$. The focal length of the combination is.......$cm$.