The correct answer is: A) Shaivism.
The Vemulawada Chalukyas were a dynasty of Indian rulers who ruled over parts of Telangana and Andhra Pradesh from the 10th to the 13th centuries. They were known for their patronage of Shaivism, and many of their temples are dedicated to Shiva.
Shaivism is one of the major Hindu traditions, and it is the dominant religion in Kashmir and parts of South India. Shaiva temples are typically dedicated to Shiva, who is one of the principal deities of Hinduism. Shiva is often depicted as a wandering ascetic, and he is associated with the destruction of the universe.
The Vemulawada Chalukyas were patrons of Shaivism, and they built many temples dedicated to Shiva. The most famous of these temples is the Vemulawada temple, which is located in the town of Vemulawada in Telangana. The temple is dedicated to Shiva, and it is one of the largest and most important Shaiva temples in South India.
The Vemulawada Chalukyas were also patrons of other religions, including Vaishnavism and Jainism. However, they were most closely associated with Shaivism, and their patronage of this tradition helped to make it one of the dominant religions in South India.