Two lenses have focal lengths $f_1$ and $f_2$ and their dispersive powers are $\omega_1$ and $\omega_2$ respectively. They will together form an achromatic combination if

$A$ convex lens of focal length $10 \, cm$ is placed in contact with a concave lens. The focal length of the combination is numerically equal to that of the concave lens. The focal length of the concave lens is ............ $cm$.

$A$ concave and a convex lens have the same focal length of $20 \ cm$ and are put into contact to form a lens combination. The combination is used to view an object of $5 \ cm$ length kept at $20 \ cm$ from the lens combination. As compared to the object,the image will be

An object is placed $30 \ cm$ away from a convex lens of focal length $10 \ cm$ and a sharp image is formed on a screen. Now, a concave lens is placed in contact with the convex lens. The screen now has to be moved by $45 \ cm$ to get a sharp image again. The magnitude of the focal length of the concave lens is (in $cm$):

$A$ collimated beam of light of diameter $2 \ mm$ is propagating along the $x$-axis. The beam is required to be expanded into a collimated beam of diameter $14 \ mm$ using a system of two convex lenses. If the first lens has a focal length of $40 \ mm$, then the focal length of the second lens is . . . . . . $mm$.

$A$ convergent doublet of separated lenses,corrected for spherical aberration,has a resultant focal length of $10\,cm$. The separation between the two lenses is $2\,cm$. The focal lengths of the component lenses are: