What does the $i$ gene encode in the $lac$ operon?

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?

Which of the following does not act as an inducer in the $lac$ operon?

In eukaryotes,the regulation of gene expression could be exerted at $:-$
$(I)$ Transcriptional level (formation of primary transcript)
$(II)$ Processing level (regulation of splicing)
$(III)$ Transport of $\text{mRNA}$ from nucleus to the cytoplasm
$(IV)$ Translational level
Choose the correct order for given statements $:-$

Select two statements out of the four given statements $(i-iv)$ with respect to the $lac$ operon.
$(i)$ Glucose or galactose binds with the repressor and inactivates it.
$(ii)$ In the absence of lactose,the repressor binds with the operator.
$(iii)$ The $Z$-gene codes for permease.
$(iv)$ It was elucidated by Francois Jacob and Jacques Monod.

In the medium where $E$. coli was growing, lactose was added, which induced the lac operon. Then, why does the lac operon shut down some time after the addition of lactose in the medium?