$\{ x \in R:|x - 2|\,\, = {x^2}\} = $

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

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^2+(a+b) x+b=0$ is necessarily true?

$I$. It has at least one negative root.

$II$. It has at least one positive root.

$III$. Both its roots are real.

If $a < 0$ then the inequality $a{x^2} - 2x + 4 > 0$ has the solution represented by

If $\alpha, \beta$ are the roots of the equation, $x^2-x-1=0$ and $S_n=2023 \alpha^n+2024 \beta^n$, then

The number of non-negative integer solutions of the equations $6 x+4 y+z=200$ and $x+y+z=100$ is