If the extreme value of $3x - 2x^2 + 1$ is $k$,then the set of all real values of $x$ for which $kx^2 + 2x + 1 > 0$ is

If $x$ is real,then the maximum and minimum values of $\frac{x^2+14x+9}{x^2+2x+3}$ are respectively

Let $x, y, z$ be positive integers such that $HCF(x, y, z)=1$ and $x^2+y^2=2z^2$. 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 $z(x^2-y^2)$.

$f(x)$ 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

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

For $x \in R$,the least value of $\frac{x^2-6x+5}{x^2+2x+1}$ is