If the bi-quadratic equation $f(x)=x^4+2x^3-16x^2-22x+7=0$ has $2+\sqrt{3}$ as one of its roots,then which of the following is not a root of $f(x)$?

Find the remainder when $x^4-11x^3+44x^2-76x+48$ is divided by $x^2-7x+12$.

Let $S = \{ \sin^2 2\theta : (\sin^4 \theta + \cos^4 \theta)x^2 + (\sin 2\theta)x + (\sin^6 \theta + \cos^6 \theta) = 0 \text{ has real roots} \}$. If $\alpha$ and $\beta$ are the smallest and largest elements of the set $S$,respectively,then $3((\alpha - 2)^2 + (\beta - 1)^2)$ equals:

The number of ordered pairs $(a, b)$ of integers such that $1 \leq a, b \leq 2021$ and the equations $x^2 - ax + b = 0$ and $x^3 - ax^2 + bx + a - b = 0$ have a common real root is

If $\alpha$ and $\beta$ $(\alpha > \beta)$ are the multiple roots of the equation $4x^4 + 4x^3 - 23x^2 - 12x + 36 = 0$,then $2\alpha - \beta = $

Let $A, G$ and $H$ be the arithmetic mean,geometric mean and harmonic mean,respectively,of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A(G-H) x^2 + G(H-A) x + H(A-G) = 0$,then: