Two thin convex lenses of focal lengths $30 \ cm$ and $10 \ cm$ are placed coaxially, $1 \ cm$ apart. The power of this combination is (in $D$):

$A$ thin convex lens of focal length $5 \ cm$ and a thin concave lens of focal length $4 \ cm$ are combined together (without any gap) and this combination has magnification $m_1$, when an object is placed $10 \ cm$ before the convex lens. Keeping the positions of the convex lens and object undisturbed, a gap of $1 \ cm$ is introduced between the lenses by moving the concave lens away, which leads to a change in the magnification of the total lens system to $m_2$. The value of $\left|\frac{m_1}{m_2}\right|$ is . . . . . . .

Two thin convex lenses are kept in contact coaxially. If the focal length of the combination of the lenses is $4 \ cm$ and the sum of the focal lengths of the two lenses is $18 \ cm$,then the focal length of the lens of lower power is (in $cm$)

The two lenses of an achromatic doublet should have:

$A$ 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?

The dispersive powers of two lenses are $0.01$ and $0.02.$ If the focal length of one lens is $+10\, cm$,what should be the focal length of the second lens so that they form an achromatic combination?