Number of solutions of equation $|x^2 -2|x||$ = $2^x$ , is

Let $p$ and $q$ be two real numbers such that $p+q=$ 3 and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to

If $\alpha ,\beta ,\gamma $are the roots of the equation ${x^3} + x + 1 = 0$, then the value of ${\alpha ^3}{\beta ^3}{\gamma ^3}$

The set of all $a \in R$ for which the equation $x | x -1|+| x +2|+a=0$ has exactly one real root is:

Let $x$ and $y$ be two $2-$digit numbers such that $y$ is obtained by reversing the digits of $x$. Suppose they also satisfy $x^2-y^2=m^2$ for some positive integer $m$. The value of $x+y+m$ is

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