Not able to login via GUI (LinuxMATE)

Namaste Friends,

Since last few days I am struggling with xinit error on my ubuntu machine.
I am not able to login via GUI but able to access my file-system via CLI.

When I try to execute <startx> it throws an error:
xinit server connection refused.

Tried to resolve the issue (related to .Xauthority file as found during
google) with <apt-get update> but failed so far.

apt-get update too throws an error:

Duplicate entries /var/lib/apt folder

Deleted sources.list file and generated new one from repogen website but
that too failed to rectify my erorr.

Is there any way out to login via GUI again. I have installed LinuxMATE
desktop and earlier it was working fine. I didn't install or update
anything which I don't understand but I don't know why all of sudden it
stopped to accept login via GUI.

Done with lots of google as according to the knowledge I have.

Kindly help if anyone has any pointers.

Thank you.

-- 

*Regards,Sahil ModGill*
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