The correct option is B 1
Two cases are possible for this equation
Case 1: 1+ex−1=ex(ex−2)
e2x−3ex=0
From this equation ex=0 which is not possible and ex=3.
Case 2: 1−ex+1=ex(ex−2)
From this equation we will get imaginary value of ex which is not possible.
So, only one solution exist.
Hence, option 'B' is correct.