If a ∈ R and the equation - 3 ( x - [ x ] ) 2 + 2 ( x - [ x ] ) + a 2 = 0 (where [ x ] denotes the greatest integer ≤ x ) has no integral solution, then all possible values of a lie in the interval

Step-by-step solution

Let x - [ x ] = t ∴    -3 t 2 + 2 t + a 2 = 0 As x is not an integer, t ≠0 ∴    a ≠0 For a root of x to exist at least one of the roots of t should be between 0 and 1 . Roots = - 2 ± 4 - 4 - 3 a 2 - 6 = - 2 ± 4 + 1 2 a 2 - 6 = - 1 ± 1 + 3 a 2 - 3 = 1 ∓ 1 + 3 a 2 3 As we observe, one root is surely less than zero, i.e., 1 - 1 + 3 a 2 3 ∴ For a solution to exist, 1 + 1 + 3 a 2 3 < 1 ∴    1 + 1 + 3 a 2 < 3 ∴&