The Arthashastra, written by Kautilya in the 4th century BCE, is an ancient Indian treatise on statecraft and economics. It includes a section on transportation and logistics, discussing the importance of efficient movement of goods and people.

Bullock carts were the primary mode of transportation in ancient India, used for both goods and people. They were sturdy and reliable, and could travel on a variety of terrains.

Brahmagupta, a 7th-century Indian mathematician, developed the concept of the 'yajamana' system for managing transportation. This system involved the use of a central authority to coordinate the movement of goods and people, ensuring efficiency and reducing congestion.

The Grand Trunk Road, built during the Mughal Empire, was primarily intended to connect major cities and facilitate trade. It stretched from Bengal in the east to Peshawar in the west, and served as a vital artery for the movement of goods and people.

Bhaskara II, an 12th-century Indian mathematician, developed the concept of 'chakravala' method for solving indeterminate equations. This method has applications in transportation optimization, as it can be used to find the shortest or most efficient routes between multiple destinations.

Camels were the primary mode of transportation for long-distance travel in ancient India. They were well-suited for desert and arid regions, and could carry heavy loads over long distances.

The concept of 'panchayati raj' system for managing transportation was not developed by any of the mentioned Indian mathematicians.

The 'sarais' built along the Grand Trunk Road were primarily intended to provide resting places for travelers. They offered shelter, food, and water to weary travelers, and also served as centers for trade and commerce.

Brahmagupta, a 7th-century Indian mathematician, developed the concept of 'kuttakara' method for solving indeterminate equations. This method has applications in transportation optimization, as it can be used to find the shortest or most efficient routes between multiple destinations.

Ships were the primary mode of transportation for water travel in ancient India. They were used for both trade and transportation of people, and could carry large quantities of goods over long distances.

Bhaskara II, an 12th-century Indian mathematician, developed the concept of 'bhaskara' method for solving indeterminate equations. This method has applications in transportation optimization, as it can be used to find the shortest or most efficient routes between multiple destinations.

The 'dharamsalas' built along the Grand Trunk Road were primarily intended to provide shelter for pilgrims. They offered free lodging and food to travelers, and also served as centers for religious and cultural activities.

Bhaskara II, an 12th-century Indian mathematician, developed the concept of 'chakravala' method for solving indeterminate equations. This method has applications in transportation optimization, as it can be used to find the shortest or most efficient routes between multiple destinations.

Air travel was not a mode of transportation in ancient India.

The concept of 'panchayati raj' system for managing transportation was not developed by any of the mentioned Indian mathematicians.