At which level is gene expression regulated in the $lac$ operon of $E. coli$?

If $E. coli$ is grown in a culture medium containing both glucose and lactose as carbon sources,when will its $lac$ operon be in the 'switch on' state?

$A$ gene that is responsible for the synthesis of a polypeptide chain is called a .......

Select the two correct statements out of the four $(i-iv)$ statements given below about the lac operon.
$(i)$ 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 Jacques Monod.
The correct statements are:

Statement $I$: Lac operator is present only in $lac$-operon and it interacts specifically with $lac$-repressor only.
Statement $II$: Each operon does not have a specific operator and a specific repressor.

The proteins encoded by all the structural genes of the $lac$ operon are essential for which process?