For what value of $\lambda$ the sum of the squares of the roots of ${x^2} + (2 + \lambda )\,x - \frac{1}{2}(1 + \lambda ) = 0$ is minimum

If $\alpha , \beta , \gamma $ are roots of equation ${x^3} + a{x^2} + bx + c = 0$, then ${\alpha ^{ - 1}} + {\beta ^{ - 1}} + {\gamma ^{ - 1}} = $

The solution of the equation $2{x^2} + 3x - 9 \le 0$ is given by

The number of solutions of the equation $x ^2+ y ^2= a ^2+ b ^2+ c ^2$. where $x , y , a , b , c$ are all prime numbers, is

The number of real solutions of the equation $|x{|^2}$-$3|x| + 2 = 0$ are

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 $..............$.