$A$ 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:

$A$ convex lens of power $+6 D$ is placed in contact with a concave lens of power $-4 D$. What is the nature and focal length of the combination?

An object $AB$ is placed $15 \text{ cm}$ on the left of a convex lens $P$ of focal length $10 \text{ cm}$. Another convex lens $Q$ is now placed $15 \text{ cm}$ to the right of lens $P$. If the focal length of lens $Q$ is $15 \text{ cm}$, the final image is . . . . . . .

An achromatic convergent doublet of two lenses in contact has a power $+2 \text{ D}$. The convex lens has power $+5 \text{ D}$. The ratio of the dispersive powers of the convergent and divergent lenses is (in magnitude)

Two lenses of focal lengths $f_1$ and $f_2$ are placed in contact,and the image is formed at a distance of $60 \, cm$. When the lenses are separated by a distance of $10 \, cm$,the image is formed at a distance of $30 \, cm$. What are the values of $f_1$ and $f_2$?

$A$ plano-convex lens of refractive index ${\mu _1}$ and focal length $f_1$ is kept in contact with another plano-concave lens of refractive index ${\mu _2}$ and focal length $f_2$. If the radius of curvature of their spherical faces is $R$ each and $f_1 = 2f_2$,then ${\mu _1}$ and ${\mu _2}$ are related as: