The least value of $\frac{x^2y^2 - 2x^2y + 2x^2 + 2xy - 2x + 1}{x^2y + x}$ is $\lambda$,where $x, y \in R^+$ and $x^2y + x \neq 0$. Then:

Let $f(x)=(x-a)(x-b)-\left(\frac{a+b}{2}\right)$. If $f(x)=0$ has both non-negative roots,then the minimum value of $f(x)$ is:

Consider the following statements: For any integer $n$,
$I.$ $n^2+3$ is never divisible by $17$.
$II.$ $n^2+4$ is never divisible by $17$.
Then,

If $x$ is real,then the difference between the greatest and least values of $\frac{x^2-x+1}{x^2+x+1}$ is

$\sqrt{x + 2\sqrt{x - 1}} + \sqrt{x - 2\sqrt{x - 1}} = $

If $\alpha$ and $\beta$ are the roots of the quadratic equation $x^2 + x \sin \theta - 2 \sin \theta = 0$,where $\theta \in (0, \frac{\pi}{2})$,then the value of $\frac{\alpha^{12} + \beta^{12}}{(\alpha^{-12} + \beta^{-12})(\alpha - \beta)^{24}}$ is equal to