The summer months are a period of excessive heat and falling air pressure in the northern half of the country.
This statement is correct. During summer, the northern part of India experiences intense heating, creating a low-pressure area. This falling air pressure is a key factor that helps draw the monsoon winds toward the Indian subcontinent.
The southwest monsoon winds are essentially 'displaced' equatorial westerlies.
This statement is correct. The southwest monsoon winds originate as equatorial westerlies that get deflected and move northeast due to the Coriolis effect when they cross the equator. They become the southwest monsoons that bring rainfall to India.
During summer the trade winds of the southern hemisphere cross the equator between 20° and 30°E longitudes.
This statement is incorrect. The southern hemisphere trade winds that contribute to the Indian monsoon typically cross the equator between 40°E and 60°E longitudes, not between 20° and 30°E. The 20°-30°E range would place them over Africa rather than the Indian Ocean region.