Among the major Indian cities listed, the city most eastward located is Option 3: Lucknow.

Lucknow, the capital city of the Indian state of Uttar Pradesh, is situated in the eastern part of the country. Here`s the detailed reasoning to support this conclusion:

1. Longitude: Longitude is the primary geographical indicator used to determine the east-west position of a location. Lucknow is located at a longitude of approximately 80.9462°E. Comparatively, Hyderabad is at around 78.4867°E, Bhopal at approximately 77.4126°E, and Bangalore at about 77.5946°E. Therefore, Lucknow has the highest longitude among the given options, indicating its position furthest eastward.

2. Uttar Pradesh`s Location: Lucknow is the capital of the state of Uttar Pradesh, which is situated in northern India. While Uttar Pradesh is predominantly located in the northern region, it extends to the eastern side of the country as well. Geographically, it borders the state of Bihar, which is known for being an eastern state in India. Therefore, Lucknow being the capital city of Uttar Pradesh aligns with its eastern positioning.

3. Neighboring States: The states bordering Lucknow and Uttar Pradesh on the east are Bihar and Jharkhand. Bihar is considered an eastern state in India and shares its eastern border with West Bengal. Jharkhand, which was previously a part of Bihar, is also situated in the eastern part of the country. The fact that Lucknow shares its border with these eastern states further supports its eastward location.

In conclusion, based on the longitude and its proximity to the eastern states of Bihar and Jharkhand, Lucknow is the most eastward located city among the options provided.