There are two white dots on a black paper separated by a distance of $1 \, mm$. They are observed by the naked eye. If the diameter of the eye lens is $3 \, mm$,what is the maximum distance between the dots and the eye so that they can be seen as separate (in $, m$)? The wavelength of light is $500 \, nm$.

When the object is self-luminous,the resolving power of a microscope is given by the expression

Given below are two statements: one is labelled as Assertion $A$ and the other is labelled as Reason $R$.
Assertion $A$: An electron microscope can achieve better resolving power than an optical microscope.
Reason $R$: The de Broglie wavelength of the electrons emitted from an electron gun is much less than the wavelength of visible light.
In the light of the above statements, choose the correct answer from the options given below:

Resolving power of a telescope can be increased by increasing

Some distant star is to be observed by a telescope with an objective lens diameter of $a$,at an angular resolution of $3.0 \times 10^{-7}$ radian. If the wavelength of light from the star reaching the telescope is $500$ nm,the minimum diameter of the objective lens of the telescope is . . . . . . cm. (nearest integer)

Assuming the human pupil to have a radius of $0.25 \ cm$ and a comfortable viewing distance of $25 \ cm$,the minimum separation between two objects that the human eye can resolve at $500 \ nm$ wavelength is nearly: (in $\mu m$)