The correct answer is option 'C', Mumbai.

The first in-house cotton mill in India was established in Mumbai, which was then known as Bombay. The establishment of this mill marked the beginning of the cotton textile industry in India.

Here is a detailed explanation:

1. Introduction:
- The cotton mill industry played a crucial role in the industrialization of India.
- It emerged during the British colonial period and became one of the major industries in the country.

2. Establishment of the First In-House Cotton Mill:
- The first in-house cotton mill in India was established in Mumbai (then Bombay) in the year 1854.
- This mill was named 'Bombay Spinning and Weaving Company'.
- It was founded by a Parsi businessman named Cowasji Nanabhai Davar.
- The mill was set up in Tardeo area of Mumbai.
- Davar was influenced by the success of textile mills in England and wanted to replicate it in India.

3. Significance of the First Mill:
- The establishment of the first in-house cotton mill in Mumbai marked the beginning of textile industrialization in India.
- It paved the way for the growth of the cotton textile industry in the country.
- The success of this mill encouraged other entrepreneurs to set up similar mills in different parts of India.

4. Growth of the Cotton Textile Industry:
- After the establishment of the first mill in Mumbai, several cotton mills were set up in other parts of the country.
- Cities like Ahmedabad, Kanpur, Coimbatore, Kolkata, and Nagpur became major centers for cotton textile production.
- The cotton textile industry became a significant contributor to India's economy and played a crucial role in employment generation.

5. Impact of the Cotton Textile Industry:
- The cotton textile industry became a major source of revenue for the British colonial government.
- It led to the growth of urban areas around the mills, as workers migrated from rural areas in search of employment opportunities.
- The industry also played a significant role in the development of infrastructure, transportation, and banking sectors.

In conclusion, the first in-house cotton mill in India was established in Mumbai (Bombay) in 1854. This marked the beginning of the cotton textile industry in the country, leading to its growth and development in various cities across India.