Option 1 states that both Statement I and Statement II are individually true, and Statement II is the correct explanation of Statement I.

Statement I states that a new hybrid architectural style developed at the beginning of the 20th century, which combined Indian and European influences. This statement is true as the architectural style known as Indo-Saracenic architecture emerged during this period and combined elements of Indian and European styles.

Statement II states that the integration of Indian and European styles in public architecture by the British was to demonstrate their legitimacy as rulers of India. This statement is also true as the British used architecture as a means to showcase their power and authority over India. They wanted to establish themselves as the rightful rulers and display their control over Indian territory.

Therefore, Option 1 is the correct answer as both statements are true, and Statement II provides the correct explanation for Statement I. No alert is needed for this answer.