If $\left|\frac{x^2+k x+1}{x^2+x+1}\right| < 3$ for all real numbers $x$,then the range of the parameter $k$ is

If $m$ is a root of the equation $(1 - ab)x^2 - (a^2 + b^2)x - (1 + ab) = 0$ and $m$ harmonic means are inserted between $a$ and $b$,then the difference between the last and the first of the means equals

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

For $0 < c < b < a$,let $(a+b-2c)x^2 + (b+c-2a)x + (c+a-2b) = 0$ and $\alpha \neq 1$ be one of its roots. Then,among the two statements:
$(I)$ If $\alpha \in (-1, 0)$,then $b$ cannot be the geometric mean of $a$ and $c$.
$(II)$ If $\alpha \in (0, 1)$,then $b$ may be the geometric mean of $a$ and $c$.

If $ax + by = 1$,where $a, b, x$ and $y$ are integers,then which one of the following is not true?

$f(x)$ is a quadratic expression such that $f(x)$ is negative when $x \in \left(-\infty, -\frac{5}{3}\right) \cup (3, \infty)$ and positive when $x \in \left(-\frac{5}{3}, 3\right)$. $g(x)$ is another quadratic expression such that $g(x)$ is negative when $x \in \left(3, \frac{9}{2}\right)$ and positive when $x \in \mathbb{R} - \left[3, \frac{9}{2}\right]$. Then,the sign of $f(x)g(x)$ in $[0, 5]$ is