Step 1: Evaluate Statement I.
Divergent evolution occurs when a common ancestral species gives rise to two or more different species.
Hence,
\[
\boxed{\text{Statement I is true}.}
\]
Step 2: Evaluate Statement II.
Analogous organs have similar functions but different origins and are evidence of
\[
\boxed{\text{Convergent evolution}.}
\]
Divergent evolution is indicated by homologous organs.
Hence,
\[
\boxed{\text{Statement II is false}.}
\]
Step 3: Choose the correct option.
Therefore,
\[
\boxed{(C)}
\]
is the correct answer.