Answer: The answer is (B). 1.01257486...
Step-by-step explanation: We are given four real numbers out of which we are to select the one which is irrational.
Option (A) is 0.12, in which the digits after the decimal are terminating. So, the number is rational.
Option (B) is 1.01257486..., the digits after the decimal are non-terminating and non-recurring. So, the number is irrational.
Option (C) is 0.1212121212..., in which the digits after the decimal are non-terminating and recurring. So, the number is rational.
Option (D) is 0.11111111..., in which the digits after the decimal are non-terminating and repeating. So, the number is rational.
Thus, (B) is the correct option.
