The number of real roots of the equation $x | x |-5| x +2|+6=0$, is

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

If $\alpha $, $\beta$, $\gamma$  are roots of ${x^3} – 2{x^2} + 3x – 2 = 0$ , then the value of$\left( {\frac{{\alpha \beta }}{{\alpha  + \beta }} + \frac{{\alpha \gamma }}{{\alpha  + \gamma }} + \frac{{\beta \gamma }}{{\beta  + \gamma }}} \right)$ is

The smallest value of ${x^2} – 3x + 3$ in the interval $( – 3,\,3/2)$ is

Let $\alpha$ and $\beta$ be the roots of $x^2-6 x-2=0$, with $\alpha>\beta$. If $a_n=\alpha^n-\beta^n$ for $n \geq 1$, then the value of $\frac{a_{10}-2 a_8}{2 a_9}$ is