Let $m$ and $n$ be the numbers of real roots of the quadratic equations $x^2-12 x+[x]+31=0$ and $x ^2-5| x +2|-4=0$ respectively, where $[ x ]$ denotes the greatest integer $\leq x$. Then $m ^2+ mn + n ^2$ is equal to $..............$.

The maximum possible number of real roots of equation ${x^5} - 6{x^2} - 4x + 5 = 0$ is

Let $P(x) = x^3 - ax^2 + bx + c$ where $a, b, c \in R$ has integral roots such that $P(6) = 3$, then $' a '$ cannot be equal to

Number of integral values of '$m$' for which $\{x\}^2 + 5m\{x\} - 3m + 1 < 0 $ $\forall x \in  R$, is (where $\{.\}$ denotes fractional part function)

The equation $x^2-4 x+[x]+3=x[x]$, where $[x]$ denotes the greatest integer function, has:

The polynomial equation $x^3-3 a x^2+\left(27 a^2+9\right) x+2016=0$ has