If two lenses of $+5$ diopters are mounted at some distance apart,the equivalent power will always be negative if the distance is

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 . . . . . . .

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

An achromatic convergent lens of focal length $20 \, cm$ is made of two lenses (in contact) of materials having dispersive powers in the ratio of $1 : 2$ and having focal lengths $f_1$ and $f_2$. Which of the following is true?

$A$ plano-convex lens of material of refractive index $\mu_1$ exactly fits into a plano-concave lens of material of refractive index $\mu_2$. If $R$ is the radius of curvature of the curved surfaces of the lenses and the plane surfaces of the lenses are parallel,the focal length of the combination is:

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?