Question

If `a in R` and the equation `-3(x-[x])^2+2(x-[x])+a^2=0` (where [x] denotes the greatest integer `le x`) has no integral solution, then all possible values of a lie in the interval: (1) (-2,-1) (2) `(oo,-2) uu (2,oo)` (3) `(-1,0) uu (0,1)` (4) (1,2)
A. `(-1,0) cup (0,1)`
B. `(1,2)`
C. `(-2,-1)`
D. `(-oo,-2)cup (2,oo)`

Answer 1

Correct Answer - A
`a^(2)=3t^(2)-2t, " where " t={x} in [0,1)`
` :. a^(2) =3t(t-2//3)`
Graph of `f(t)=3t(t-2//3)` is as shown in the following figure.

Clearly, from graph
`0 lt a^(2) lt 1`
`or a in (-1,0) cup (0,1) `