An object of height $25 \ cm$ is placed in front of a convex lens of focal length $30 \ cm$. If the height of the formed image is $50 \ cm$,then the distance between the object and the image is ..... $cm$.

Consider the following statements regarding the real images formed with a converging lens.
$I$. Real images can be seen only if the image is projected onto the screen.
$II$. The real image can be seen only from the same side of the lens as that on which the object is positioned.
$III$. Real images produced by converging lenses are not only laterally but also longitudinally inverted as with mirrors.
Which of the above statement$(s)$ is/are incorrect?

$A$ bi-concave symmetric lens made of glass has a refractive index of $1.5$. It has both surfaces with the same radius of curvature $R$. On immersion in a liquid of refractive index $1.75$,it will behave as a:

The dispersive powers of the glasses of lenses used in an achromatic pair are in the ratio $5 : 3$. If the focal length of the concave lens is $15 \ cm$,then the nature and focal length of the other lens would be:

The power of a biconvex lens is $1.25\,m^{-1}$ in a particular medium. The refractive index of the lens is $1.5$ and the radii of curvature are $20\,cm$ and $40\,cm$ respectively. Find the refractive index of the surrounding medium.

Explain how the image is formed by a thin convex lens and derive the lens maker's formula: $\frac{1}{f} = (n_{21} - 1) \left[ \frac{1}{R_1} - \frac{1}{R_2} \right]$.