The question asks about the first female Finance Minister of independent India. Let's analyze the given options:
- Indira Gandhi: Indira Gandhi, a prominent political figure, served as Prime Minister of India. She also held the Finance Ministry portfolio.
- Sarojini Naidu: Known as the "Nightingale of India," Naidu was an activist and poet, but never served as Finance Minister.
- Vijaya Lakshmi Pandit: She was a diplomat and the first woman to become President of the United Nations General Assembly, but did not serve as Finance Minister.
- Rajkumari Amrit Kaur: She was the first Health Minister after independence and a notable freedom fighter, but not involved with the Finance Ministry.
From these options, Indira Gandhi is the only individual to have served as India's Finance Minister. Therefore, the correct answer is:
Indira Gandhi