A real root of the equation ${\log _4}\{ {\log _2}(\sqrt {x + 8} - \sqrt x )\} = 0$ is

In a cubic equation coefficient of $x^2$ is zero and remaining coefficient are real has one root $\alpha = 3 + 4\, i$ and remaining roots are $\beta$ and $\gamma$ then $\alpha \beta \gamma$ is :-

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 two roots of an equation ${x^3} - 9{x^2} + 14x + 24 = 0$ are in the ratio $3 : 2$. The roots will be

The sum of all integral values of $\mathrm{k}(\mathrm{k} \neq 0$ ) for which the equation $\frac{2}{x-1}-\frac{1}{x-2}=\frac{2}{k}$ in $x$ has no real roots, is ..... .

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