Concept:
To find the odd one out, identify the common property among the places.
Step 1: Observe the cities.
Most of the given cities are state capitals:
\[
\text{Bengaluru} \rightarrow \text{Karnataka}
\]
\[
\text{Chennai} \rightarrow \text{Tamil Nadu}
\]
\[
\text{Mumbai} \rightarrow \text{Maharashtra}
\]
\[
\text{Patna} \rightarrow \text{Bihar}
\]
\[
\text{Lucknow} \rightarrow \text{Uttar Pradesh}
\]
\[
\text{Jaipur} \rightarrow \text{Rajasthan}
\]
\[
\text{Bhuvaneshwar} \rightarrow \text{Odisha}
\]
\[
\text{Trivandrum} \rightarrow \text{Kerala}
\]
Step 2: Check Mangalore.
Mangalore is an important city in Karnataka, but it is not a state capital.
Thus, it does not belong to the group.
Hence, the odd one out is:
\[
\boxed{\text{Mangalore}}
\]