$A$ thin convex lens of focal length $10 \ cm$ is placed in contact with a concave lens of the same material and of the same focal length. The focal length of the combination will be:

The ratio of the dispersive powers of two lenses is $4:3$. If the achromatic combination of these two lenses in contact is a convex lens of focal length $60 \, cm$,find the focal lengths of the component lenses.

An achromatic convergent doublet of two lenses in contact has a power of $+2 \ D$. The convex lens has a power of $+5 \ D$. What is the ratio of the dispersive powers of the convergent and divergent lenses?

$A$ combination of two thin convex lenses of focal lengths $0.3 \, m$ and $0.1 \, m$ will have minimum spherical and chromatic aberrations if the distance between them is . . . . . . $m$.

Two plano-convex lenses,each of focal length $10 \, cm$ and refractive index $3/2$,are placed as shown in the figure. The space between them is filled with water $\left( \mu = 4/3 \right)$. The whole arrangement is in air. The optical power of the system is ....... $D$.

$A$ point object,$O$ is placed in front of two thin symmetrical coaxial convex lenses $L_1$ and $L_2$ with focal lengths $24\,cm$ and $9\,cm$ respectively. The distance between the two lenses is $10\,cm$ and the object is placed $6\,cm$ away from lens $L_1$ as shown in the figure. The distance between the object and the image formed by the system of two lenses is .........$cm$.