If $x$ is real, then the value of ${x^2} - 6x + 13$ will not be less than

Let $x, y, z$ be positive integers such that $HCF$ $(x, y, z)=1$ and $x^2+y^2=2 z^2$. Which of the following statements are true?

$I$. $4$ divides $x$ or $4$ divides $y$.

$II$. $3$ divides $x+y$ or $3$ divides $x-y$.

$III$. $5$ divides $z\left(x^2-y^2\right)$.

Let $x, y, z$ be non-zero real numbers such that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=7$ and $\frac{y}{x}+\frac{z}{y}+\frac{x}{z}=9$, then $\frac{x^3}{y^3}+\frac{y^3}{z^3}+\frac{z^3}{x^3}-3$ is equal to

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

If $\alpha , \beta $ are the roots of the equation $x^2 - 2x + 4 = 0$ , then the value of $\alpha ^n +\beta ^n$ is

If $\alpha,\beta,\gamma, \delta$ are the roots of $x^4-100x^3+2x^2+4x+10 = 0$ then $\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}$ is equal to :-