Question

If $a\in R$ and the equation $-3(x-[x]{)}^2+2(x-\left[x\right])+a^2=0$ (where $\left[x\right]$ denotes the greatest integer $\le x$ ) has no integral solution but has real solutions, then all possible values of $a$ lie in the interval:

• A. $(-1,0)\cup (0,1)$
• B. $(1,2)$
• C. $(-2,-1)$
• D. $(-\infty ,-2)\cup (2,\infty )$

Answer 1

Answer: (A)

$(-1,0)\cup (0,1)$

$a^2=3t^2-2t$
For non-integral solution,
$0<a^2<1$
$a\in (-1,0)\cup (0,1)$
[Note: It is assumed that a real solution of given equation exists].