E.coli cells with a mustard $z$ gene of the lac operon cannot grow in medium containing only lactose as the…
E.coli cells with a mustard $z$ gene of the lac operon cannot grow in medium containing only lactose as the source of energy because:
the lac operon is constitutively active in these cells
they cannot synthesize functional $\beta$ galactosidase
in the presence of glucose, E.coli cells do not utilize lactose
they cannot transport lactose from the medium into the cell.
Solution
The $z$ gene in the lac operon encodes for a $\beta$ galactosidase protein. This protein is an enzyme that is capable of breaking down the lactose into simpler sugars which can be absorbed by the bacteria. Hence a mutated $z$ gene would result in a non-functional enzyme and thus the bacteria cannot grow.
Related Theory
Glucose is the preferred and most frequently available energy source for E. coli. The enzymes to metabolize glucose are made constantly by E. coli. When E. coli is exposed to both lactose and glucose, the organism first metabolise glucose and then temporarily cease to grow until the genes of lac operon become included to provide the ability to metabolise lactose.