The correct answer is option 'C' - 1 and 2. The factors that influence the density distribution of population in India most profoundly are the amount of rainfall and the fertility of soils. Let's delve into the details:

1. Amount of rainfall:
- Rainfall is a critical factor that determines the availability of water for agriculture, which is the primary occupation for a large portion of the Indian population.
- Regions with adequate rainfall are more suitable for agriculture, leading to higher population density as people settle in these areas to cultivate crops.
- In contrast, areas with low rainfall face water scarcity and are less suitable for agriculture, resulting in lower population density.

2. Fertility of soils:
- Fertile soils play a crucial role in determining the agricultural productivity of a region.
- Areas with fertile soils provide better conditions for crop growth and higher yields, attracting agricultural activities and settlements.
- The fertility of soils is influenced by factors such as the presence of organic matter, nutrient content, and soil composition, which vary across different regions of India.
- Regions with highly fertile soils tend to have higher population density as they offer better agricultural opportunities.

Other factors mentioned in the options, such as the distribution of minerals and cultural factors, do have some influence on the density distribution of population in India but not as profoundly as rainfall and soil fertility.

3. Distribution of minerals:
- The distribution of minerals has an impact on economic activities and industrial development in specific regions.
- However, mineral-rich areas may not necessarily have a high population density as mineral extraction activities often require fewer workers compared to agriculture.
- Moreover, mineral-rich areas may face challenges in terms of accessibility, infrastructure, and environmental concerns, limiting their potential for settlement and population growth.

4. Cultural factors:
- Cultural factors, such as language, religion, and traditions, can influence the settlement patterns and distribution of population to some extent.
- However, cultural factors alone are not the most significant determinants of population density.
- Cultural factors may play a role in shaping regional preferences for settlement but are secondary to environmental factors like rainfall and soil fertility.

In conclusion, while factors like the distribution of minerals and cultural factors do have some influence on the density distribution of population in India, the amount of rainfall and the fertility of soils are the most profound factors. These environmental factors directly affect agricultural productivity and determine the suitability of regions for settlement and agriculture, thereby playing a crucial role in shaping population distribution in the country.