**Explanation:**

The correct answer is option 'A': Barmer and Jaisalmer districts.

**Petroleum discovery in Rajasthan:**

Petroleum or crude oil is a naturally occurring fossil fuel found beneath the Earth's surface. It is an essential energy resource used for various purposes like transportation, electricity generation, and industrial processes.

Rajasthan, the largest state in India, is not traditionally known for its oil reserves. However, significant petroleum reserves have been discovered in the western part of the state, particularly in the districts of Barmer and Jaisalmer.

**1. Barmer district:**

Barmer district is located in the western part of Rajasthan. It shares its borders with Pakistan and the Indian state of Gujarat. The petroleum reserves in Barmer were discovered in the Mangala field, which is a part of the larger Barmer-Sanchor Basin. The Mangala field is operated by Cairn India, which is a subsidiary of Vedanta Resources. The oilfields in Barmer have been estimated to hold significant reserves, making them one of the largest onshore oil discoveries in India.

**2. Jaisalmer district:**

Jaisalmer district is situated in the westernmost part of Rajasthan, near the international border with Pakistan. The petroleum reserves in Jaisalmer were discovered in the Fatehgarh field. The Fatehgarh field is operated by Oil and Natural Gas Corporation (ONGC), which is a state-owned oil and gas company in India. The oilfields in Jaisalmer have also been found to hold significant reserves.

**Significance of the discoveries:**

The discoveries of petroleum in Barmer and Jaisalmer districts have brought about several significant economic and developmental implications for the state of Rajasthan and the country as a whole.

1. Economic growth: The presence of significant petroleum reserves in Rajasthan has opened up new avenues for economic growth. The extraction and production of oil have created job opportunities and contributed to the state's revenue through taxes and royalties.

2. Energy security: India, being a rapidly growing economy, has a high demand for energy resources. The petroleum discoveries in Rajasthan have helped enhance the country's energy security by reducing its dependence on imported oil.

3. Infrastructural development: The petroleum discoveries have led to the development of infrastructure in the region. This includes the construction of pipelines, refineries, storage facilities, and transportation networks, which are essential for the extraction, processing, and distribution of petroleum products.

4. Foreign investment: The petroleum discoveries have attracted foreign investment in the oil and gas sector. Companies like Cairn India (Vedanta Resources) and ONGC have invested significant capital in exploring and developing the oilfields in Barmer and Jaisalmer.

In conclusion, the districts of Barmer and Jaisalmer in Rajasthan have witnessed significant petroleum discoveries, which have brought about economic growth, energy security, infrastructural development, and foreign investment in the state. These discoveries have contributed to the overall development and progress of Rajasthan and the country as a whole.

Since these are located in the sedimentary basin.