Invest India is the investment promotion and facilitation agency of India. It is responsible for promoting and facilitating investment in the country. Let's analyze the given statements to determine their correctness.

Statement 1: It is the investment promotion and facilitation agency of India.
This statement is correct. Invest India is indeed the investment promotion and facilitation agency of India. It was established in 2009 as a non-profit venture under the Department for Promotion of Industry and Internal Trade (DPIIT), Ministry of Commerce and Industry.

Statement 2: It is a public entity with equal shareholdings between the center and state governments.
This statement is not correct. Invest India is not a public entity with equal shareholdings between the center and state governments. It is a non-profit venture and does not have any shareholdings or ownership structure involving the center and state governments.

Therefore, only statement 1 is correct.

Invest India plays a crucial role in promoting India as an investment destination and providing guidance and facilitation to both domestic and international investors. It acts as a one-stop solution for investors, providing them with all the necessary information, guidance, and support to establish and expand their businesses in India.

The agency works closely with various government departments and agencies at the central and state levels to address investor queries, provide investment-related information, and facilitate investment projects. It also helps in identifying investment opportunities, conducting investment promotion activities, and organizing investor outreach programs.

Invest India has been instrumental in implementing various initiatives and reforms to improve the ease of doing business in India. It provides assistance in obtaining various regulatory approvals, licenses, and permits required for setting up and operating businesses in the country. It also helps investors navigate the bureaucratic processes and overcome any challenges they may face during the investment process.

Overall, Invest India plays a crucial role in attracting and facilitating investment in India. It acts as a bridge between investors and the Indian government, providing a conducive environment for investment and contributing to the country's economic growth and development.