Assertion: Towards the beginning of the twentieth century, a new hybrid architectural style developed which combined the Indian with the European. This was called the Eurasian style. 
Reason:  By integrating Indian and European styles in public architecture the British wanted to prove that they were legitimate rulers of India.

The Assertion is incorrect but the Reason is correct.

The correct answer is Option 4 - The Assertion is incorrect but the Reason is correct.

Assertion: Towards the beginning of the twentieth century, a new hybrid architectural style developed which combined the Indian with the European. This was called the Eurasian style. (INCORRECT)
Reason:  By integrating Indian and European styles in public architecture the British wanted to prove that they were legitimate rulers of India. (CORRECT)

CORRECTION in Assertion: Towards the beginning of the twentieth century, a new hybrid architectural style developed which combined the Indian with the European. This was called Indo-Saracenic.

Towards the beginning of the twentieth century, a new hybrid architectural style developed which combined the Indian with the European. This was called Indo-Saracenic.
'Indo' was shorthand for Hindu and 'Saracen' was a term Europeans used to designate Muslim. The inspiration for this style was medieval buildings in India, with domes, chhatris, jalis, and arches.
By integrating Indian and European styles in public architecture the British wanted to prove that they were legitimate rulers of India.
The Gateway of India, built in the traditional Gujarati style to welcome King George V and Queen Mary to India in 1911, is the most famous example of this style.
The industrialist Jamsetji Tata Built the Taj Mahal Hotel in a similar style. Besides being a symbol of Indian enterprise, the building became a challenge to the racially exclusive clubs and hotels maintained by the British.