Let $a \in \mathbb{R}$ and let $\alpha, \beta$ be the roots of the equation $x^2+60^{\frac{1}{4}} x+a=0$. If $\alpha^4+\beta^4=-30$,then the product of all possible values of $a$ is $......$

Let $\alpha_\theta$ and $\beta_\theta$ be the distinct roots of $2x^2 + (\cos \theta)x - 1 = 0$,where $\theta \in (0, 2\pi)$. If $m$ and $M$ are the minimum and the maximum values of $\alpha_\theta^4 + \beta_\theta^4$,then $16(M + m)$ equals:

If $\alpha$ is a root of the quadratic equation $x^2 + 6x - 2 = 0$,then another root $\beta$ is

Let $(a-3)x^2+12x+(a+6)>0, \forall x \in R$ and $a \in (\ell, \infty)$. If $\alpha$ is the least positive integral value of $a$,then the roots of $(\alpha-3)x^2+12x+(\ell+2)=0$ are

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

If $a, b, c$ are real and $x^3 - 3b^2x + 2c^3$ is divisible by $x - a$ and $x - b$,then: