The number of real solutions of the equation $x\left(x^2+3|x|+5|x-1|+6|x-2|\right)=0$ is

The set of all real numbers $x$ for which ${x^2} - |x + 2| + x > 0,$ is

If for a posiive integer $n$ , the quadratic equation, $x\left( {x + 1} \right) + \left( {x + 1} \right)\left( {x + 2} \right) + .\;.\;.\; + \left( {x + \overline {n - 1} } \right)\left( {x + n} \right) = 10n$ has two consecutive integral solutions, then $n$ is equal to:

The equation${e^x} - x - 1 = 0$ has

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.

The sum of the roots of the equation, ${x^2}\, + \,\left| {2x - 3} \right|\, - \,4\, = \,0,$ is