Correct Option (2)

The coding scheme follows a pattern where the input number, when expressed as a cube of an integer (x³), is transformed into a coded number by cubing the base increased by 3, i.e., (x+3)³.

For the input 64: 64 is 4³. Applying the rule, the coded number is (4+3)³ = 7³ = 343.
For the input 216: 216 is 6³. Applying the rule, the coded number is (6+3)³ = 9³ = 729.
Following this established pattern for the input 512: 512 is 8³. The coded number will be (8+3)³ = 11³ = 1331.

Incorrect Options:

The other options do not adhere to the identified coding pattern, which requires the base of the cube to be incremented by exactly 3.

1) 1000: This is 10³. If 512 (8³) were coded as 1000, the increment in the base would be 2 (10-8), which is inconsistent with the established pattern of an increment of 3.
3) 1728: This is 12³. If 512 (8³) were coded as 1728, the increment in the base would be 4 (12-8), which is inconsistent with the established pattern of an increment of 3.
4) 2197: This is 13³. If 512 (8³) were coded as 2197, the increment in the base would be 5 (13-8), which is inconsistent with the established pattern of an increment of 3.