For real values of $x$ and $a$,if the expression $\frac{x+a}{2 x^2-3 x+1}$ assumes all real values,then

If $\alpha$ and $\beta$ are the roots of the equation $ax^2 + bx + c = 0$,and $\alpha + \beta$,$\alpha^2 + \beta^2$,and $\alpha^3 + \beta^3$ are in a geometric progression,and $\Delta = b^2 - 4ac$,then which of the following is true?

If $\frac{x-P}{x^2-3x+2}$ takes all real values for $x \in \mathbb{R} \setminus \{1, 2\}$,then the range of $P$ is

If the equation $x^8 - kx^2 + 3 = 0$ has a real solution,then the least integral value of $k$ is-

Let $p$ and $q$ be two real numbers such that $p+q=3$ and $p^{4}+q^{4}=369$. Then $\left(\frac{1}{p}+\frac{1}{q}\right)^{-2}$ is equal to

$f(x)$ 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