Unpolarised light falls directly on a half-wave plate Will an unpolarised light get polarised if it falls directly on a half-wave plate? If so, then what will be the state of polarisation? 
 A: If you shine unpolarized light onto a half-wave plate, then the result will be similarly unpolarized.
It seems you really don't want to believe this answer, so let's address your two objections thus far:


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But.... By the definition of half wave plate.... It says that an unpolarised light will be divided into two polarised lights... E-ray and o-ray.... And a phase difference of (π) will be there 

Yes. But in unpolarized light, those two components are incoherent with respect to each other, and they don't have a well-defined relative phase. Adding $\pi$ phase to that undefined relative phase doesn't change anything.

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But I found half wave plate will rotate plane of polarisation by 2theta....so won't it be 90degree rotation?

First of all, half-wave plates don't rotate the plane of polarization - they reflect it. (To rotate the polarization, you need two subsequent half-wave plates, with their slow axes at an angle.) In any case, if you want to analyze it like that, then the unpolarized light is the incoherent mixture of two orthogonal polarizations, which will be taken by the HWP's reflection to two different orthogonal polarizations, still with no relative coherence. Or, in other words, unpolarized light.
