Step 1: Conceptual Understanding:
The objective is to associate prominent modern Indian art organizations and institutions with their founding dates or periods. Certain years provided in the options may be approximate.
Step 3: Detailed Analysis:
(A) Calcutta Group: Established in 1943, it is one of India's early modern art collectives. Thus, (A) corresponds to (IV).
(B) The Progressive Artists' Group (PAG): Founded in Bombay (now Mumbai) in 1947, PAG is a significant artist group. The closest option is 1946, likely an approximation for the question. Therefore, (B) corresponds to (III).
(C) Group 1890: Formed in 1962, this group held its sole exhibition in 1963. The exhibition year is typically considered its definitive date. Hence, (C) corresponds to (II).
(D) Baroda Art School: The Faculty of Fine Arts at Maharaja Sayajirao University of Baroda was established in 1950. The option "Late 1950s" best reflects the period of its significant growth and the formation of its distinct identity, making it the most appropriate general match. Consequently, (D) corresponds to (I).
Step 4: Conclusion:
The optimal matching is (A) - (IV), (B) - (III), (C) - (II), and (D) - (I).