Let $S=\left\{\sin ^2 2 \theta:\left(\sin ^4 \theta+\cos ^4 \theta\right) x^2+(\sin 2 \theta) x+\right.$ $\left(\sin ^6 \theta+\cos ^6 \theta\right)=0$ has real roots $\}$. If $\alpha$ and $\beta$ be the smallest and largest elements of the set $S$, respectively, then $3\left((\alpha-2)^2+(\beta-1)^2\right)$ equals....................
$4$
$2$
$7$
$9$
The set of all $a \in R$ for which the equation $x | x -1|+| x +2|+a=0$ has exactly one real root is:
If $\log _{(3 x-1)}(x-2)=\log _{\left(9 x^2-6 x+1\right)}\left(2 x^2-10 x-2\right)$, then $x$ equals
$\{ x \in R:|x - 2|\,\, = {x^2}\} = $
Sum of the solutions of the equation $\left[ {{x^2}} \right] - 2x + 1 = 0$ is (where $[.]$ denotes greatest integer function)
The real roots of the equation ${x^2} + 5|x| + \,\,4 = 0$ are