If $x = \frac{1}{2} \left( \sqrt{7} + \frac{1}{\sqrt{7}} \right)$,then the value of $\frac{\sqrt{x^2 - 1}}{x - \sqrt{x^2 - 1}}$ is equal to

The minimum value of $f(x) = \frac{x^2-2x+3}{x^2-4x+7}$ is

If $a, b, c \in \mathbb{R}$ are such that $4a + 2b + c > 0$ and $ax^2 + bx + c = 0$ has no real roots,then the value of $(c + a)(c + b)$ is

Let $\alpha, \beta \in \mathbb{N}$ be roots of the equation $x^2-70x+\lambda=0$,where $\frac{\lambda}{2}, \frac{\lambda}{3} \notin \mathbb{N}$. If $\lambda$ assumes the minimum possible value,then $\frac{(\sqrt{\alpha-1}+\sqrt{\beta-1})(\lambda+35)}{|\alpha-\beta|}$ is equal to :

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,the value of $\frac{x^3}{y^3}+\frac{y^3}{z^3}+\frac{z^3}{x^3}-3$ is equal to

For all real values of $x$, the minimum value of $\frac{1-x+x^2}{1+x+x^2}$ is