Option 1: Balban - Balban was a Delhi Sultan, but there is no historical evidence to suggest that he raised the height of the Qutub Minar.

Option 2: Alauddin Khalji - Alauddin Khalji was also a Delhi Sultan, but there is no historical evidence to suggest that he raised the height of the Qutub Minar either.

Option 3: Muhammad Tughlaq - Muhammad Tughlaq was a Delhi Sultan known for his architectural projects, but there is no historical evidence to suggest that he raised the height of the Qutub Minar.

Option 4: Firoz Shah Tughlaq - Firoz Shah Tughlaq was a Delhi Sultan who ruled in the 14th century. He is credited with raising the height of the Qutub Minar by adding the fifth storey. Historical records indicate that he undertook the restoration and renovation of several monuments, including the Qutub Minar.

Therefore, the correct answer is option 4 - Firoz Shah Tughlaq. He is the Delhi Sultan who raised the height of the Qutub Minar by adding the fifth storey.