Explanation:

It was during the reign of Iltutmish that Chengiz Khan reached the Indus in pursuit of the fugitive Khwarezm prince.

This statement is correct. During the reign of Iltutmish (r. 1211–1236 CE), Chengiz Khan invaded Central Asia and pursued the Khwarezm Shah to the region near the Indus River. However, Iltutmish wisely chose not to engage with the Mongol leader, ensuring that India was not invaded by the Mongols at this time.

It was during the reign of Muhammad bin Tughluq that Taimur occupied Multan and crossed the Indus.

This statement is incorrect. Taimur's invasion of India took place in 1398, during the reign of Nasir-ud-Din Mahmud Shah Tughluq, a later ruler of the Tughluq dynasty, and not during the reign of Muhammad bin Tughluq.

It was during the reign of Deva Raya II of the Vijayanagara Empire that Vasco da Gama reached the coast of Kerala.

This statement is incorrect. Vasco da Gama arrived at the coast of Kerala in 1498, specifically at Calicut. By that time, Deva Raya II (r. 1424–1446 CE) had already passed, and the ruler of the Vijayanagara Empire at the time of da Gama's arrival was not Deva Raya II.

Given the evaluations, the correct statement is only the first one. Hence, the correct answer is Option 1.