Final answer:
A mutation leading to 10-fold greater transcription of the cII gene in phage λ would favor the lysogenic cycle, where the phage DNA integrates with the host genome, providing an evolutionary advantage to the virus.
Step-by-step explanation:
If a mutation occurs in phage λ resulting in 10-fold greater transcription of the cII gene, the lysogenic cycle would be favored. The cII gene is a key regulator that, when highly expressed, leads to the integration of the phage genome into the bacterial host genome. This is a step in the lysogenic cycle where the phage DNA becomes a prophage and replicates along with the host cell's genome. This mode of replication can provide an evolutionary advantage as it allows the virus to persist within the host without killing it, and under stress conditions, the prophage can be excised to enter the lytic cycle, where the phage replicates and eventually lyses the host cell to release new virions.