Which of the following is the substrate for $\beta$-galactosidase?

Which of the following statements about the $lac$ operon is correct?

In the given diagram of the lac operon,which one is the inducer?

Where are the regulatory genes located in an operon?

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

An organism is able to survive on a culture medium containing nutrient $A$ through an enzyme-catalyzed process: $A \xrightarrow{X} B \xrightarrow{Y} C$. $A$ mutant organism fails to survive on this medium. However,it grows when nutrient $B$ is added to the medium. Which gene of this mutant is defective?