An object placed at a distance of $16 \ cm$ from a convex lens produces an image of magnification $m$ $(m > 1)$. If the object is moved towards the lens by $8 \ cm$, then again an image of magnification $m$ is obtained. The numerical value of the focal length of the lens is (in $cm$)

$A$ real object approaches a fixed diverging lens with a constant velocity from infinity along the principal axis. The relative velocity between the object and its image will be

$A$ plane wave passes through a convex lens. The geometrical shape of the wavefront that emerges is

Remote sensing satellites move in an orbit that is at an average height of about $500 \, km$ from the surface of the earth. The camera onboard one such satellite has a screen of area $A$ on which the images captured by it are formed. If the focal length of the camera lens is $50 \, cm$,then the terrestrial area that can be observed from the satellite is close to ............... $A$.

$A$ thin convex lens is made of two materials with refractive indices $n_1$ and $n_2$,as shown in the figure. The radii of curvature of the left and right spherical surfaces are equal. $f$ is the focal length of the lens when $n_1 = n_2 = n$. The focal length is $f + \Delta f$ when $n_1 = n$ and $n_2 = n + \Delta n$. Assuming $\Delta n \ll (n - 1)$ and $1 < n < 2$,which of the following statement$(s)$ is/are correct?
$(1)$ The relation between $\frac{\Delta f}{f}$ and $\frac{\Delta n}{n}$ remains unchanged if both the convex surfaces are replaced by concave surfaces of the same radius of curvature.
$(2)$ $\left|\frac{\Delta f}{f}\right| < \left|\frac{\Delta n}{n}\right|$
$(3)$ For $n = 1.5, \Delta n = 10^{-3}$ and $f = 20 \text{ cm}$,the value of $|\Delta f|$ will be $0.04 \text{ cm}$.
$(4)$ If $\frac{\Delta n}{n} < 0$ then $\frac{\Delta f}{f} > 0$.

$A$ hollow double concave lens is made of very thin transparent material. It can be filled with air or either of two liquids $L_1$ and $L_2$ having refractive indices $n_1$ and $n_2$ respectively $(n_2 > n_1 > 1)$. The lens will diverge a parallel beam of light if it is filled with