$i.$ Glucose or galactose may bind with the repressor and inactivate it.
$ii.$ In the absence of lactose, the repressor binds with the operator region.
$iii.$ The z-gene codes for permease.
$iv.$ This was elucidated by Francois Jacob and Jacque Monod.
Reason : $DNA$ binds around histone proteins that form a pool and the entire structure is called a nucleosome.