Large apertures of telescopes are used for

Which of the following is the smallest size grain that can be seen using a normal microscope using visible light $(400 \ nm < \lambda < 700 \ nm)$ (in $nm$)?

An astronaut is looking down on the Earth's surface from a space shuttle at an altitude of $400 \, km$. Assuming that the astronaut's pupil diameter is $5 \, mm$ and the wavelength of visible light is $500 \, nm$,the astronaut will be able to resolve a linear object of the size of about ........ $m$.

The diameter of the objective of a telescope is $200 \text{ cm}$. What is the resolving power of the telescope? Take the wavelength of light $\lambda = 5000 \text{ \AA}$.

Two-point white dots are $2 \,mm$ apart on a black paper. They are viewed by an eye of pupil diameter $3 \,mm$. What is the maximum distance at which these dots can be resolved by the eye (in $\,m$)? $(\lambda = 500 \,nm)$

We use a simple microscope to magnify an object. The microscope has a numerical aperture of $\sin \alpha = 0.24$. The object is so small that the resolving power of the microscope is fully utilized. If the diameter of the eye's pupil is $d = 4.0 \ mm$ and the least distance of distinct vision is $D = 25 \ cm$,what is the minimum magnifying power of the microscope?