The correct option is A (−1,0)∪(0,1)
Put t = x – [x] = {X}, which is a fractional part function and lie between 0≤x<1 and then solve it.
Given, a∈R and equation is −3{x−[x]}2+2{x−[x]}+a2=0
Let t = x – [x], then equation is
−3t2+2t+a2=0⇒t=1±√1+3a23∵t=x−[x]=x[fractionalpart]∴0≤t≤1
0≤1+√1+3a23<1[∵Fractionalpart(x)>0]⇒√1+3a2<2⇒1+3a2<4⇒a2−1<0⇒(a+1)(a−1)<0
∴ a lies in (-1,1), for no integer solution we consider (−1,0)∪(0,1)