$A$ thin biconvex lens is prepared from glass $(mu = 1.5)$ where both curved surfaces have equal radii of $20 \text{ cm}$ each. The left side surface of the lens is silvered from the outside to make it reflecting. To have the position of the image and the object at the same place, the object should be placed from the lens at a distance of . . . . . . $\text{cm}$.

The figure shows an equiconvex lens of focal length $f$. If the lens is cut along $AB$,the focal length of each half will be

$A$ symmetric double convex lens is cut into two equal parts by a plane perpendicular to the principal axis. If the power of the original lens was $4 \ D$,what will be the power of each cut lens (in $D$)?

Consider a concave mirror and a convex lens (refractive index $=1.5$) of focal length $10 \ cm$ each,separated by a distance of $50 \ cm$ in air (refractive index $=1$) as shown in the figure. An object is placed at a distance of $15 \ cm$ from the mirror. Its erect image formed by this combination has magnification $M_1$. When the setup is kept in a medium of refractive index $7/6$,the magnification becomes $M_2$. The magnitude $\left|\frac{M_2}{M_1}\right|$ is

$A$ point source is placed $20 \,cm$ to the left of a concave lens of focal length $10 \,cm$.
$(a)$ Where is the image formed?
$(b)$ Where to the right of the lens would you place a concave mirror of focal length $5 \,cm$,so that the final image is coincident with the source?
$(c)$ Where would the final image be formed,if the concave mirror is replaced by a plane mirror at the same position?

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 .........