The focal length of a lens of dispersive power $0.45$ which should be placed in contact with a convex lens of focal length $84 \,cm$ and dispersive power $0.21$ to make the achromatic combination from the two lenses,in $cm$ is

As shown in the figure, a combination of a thin plano-concave lens and a thin plano-convex lens is used to image an object placed at infinity. The radius of curvature of both the lenses is $30\,cm$ and the refractive index of the material for both the lenses is $1.75$. Both the lenses are placed at a distance of $40\,cm$ from each other. Due to the combination, the image of the object is formed at a distance $x = ..........cm$ from the concave lens.

Concave and convex lenses are placed touching each other. The ratio of the magnitudes of their power is $2:3$. The focal length of the system is $30 \ cm$. Then the focal lengths of the individual lenses are:

$A$ convex lens of focal length $0.5 \ m$ and a concave lens of focal length $1 \ m$ are combined. The power of the resulting lens will be: (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 . . . . . . .

$A$ convex lens $A$ of focal length $20 \, cm$ and a concave lens $B$ of focal length $5 \, cm$ are kept along the same axis with a distance $d$ between them. If a parallel beam of light falling on $A$ leaves $B$ as a parallel beam,then the distance $d$ in $cm$ will be: