Which of the following research institutes were established between the first and third five-year plans?

(A) Indian Institute of Science

(B) Saha Institute of Nuclear physics.

(C) Lok Bharati

(D) Gandhi Gram Rural Institute

(E) Raman Research Institute

Choose the correct answer from the options given below:

The correct option is (B), (C) and (D) Only.

To understand why this option is correct, let us look at the timeline of the first three Five-Year Plans of India and the establishment years of each of the research institutes listed in the question:
The First Five-Year Plan spanned from 1951 to 1956.
The Second Five-Year Plan spanned from 1956 to 1961.
The Third Five-Year Plan spanned from 1961 to 1966.
Therefore, the period between the beginning of the first and the end of the third five-year plans is from 1951 to 1966.

Now, let us examine the establishment year of each institute:
1. Indian Institute of Science (A): Established in the year 1909. Since 1909 is before 1951, it does not fall within the specified period.
2. Saha Institute of Nuclear Physics (B): Founded by Meghnad Saha, it was formally inaugurated in 1950, and its transit building was established and opened in 1951. Thus, it falls within the first five-year plan period.
3. Lok Bharati (C): Established in the year 1953 by Nanabhai Bhatt in Sanosara, Gujarat, as a premier rural education institute. This falls within the first five-year plan period (1951–1956).
4. Gandhi Gram Rural Institute (D): Founded in 1956 by Dr. G. Ramachandran and Dr. Soundaram Ramachandran to implement Mahatma Gandhi's Nai Talim system of education. It falls right at the transition of the first and second five-year plans, within the specified duration.
5. Raman Research Institute (E): Founded by Nobel laureate Sir C. V. Raman in the year 1948, which is prior to the commencement of the first five-year plan in 1951.

Based on these timeline matches, only the institutes (B) Saha Institute of Nuclear Physics, (C) Lok Bharati, and (D) Gandhi Gram Rural Institute were established between the first and third five-year plans (1951–1966).