The equation of lowest degree with rational coefficients having roots $\sqrt{3}+\sqrt{2} i$ and $\sqrt{3}-\sqrt{2}$ 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 $c$.

$p$ and $q$ are two roots of the equation $x^2+7x+3=0$. If $\frac{3p}{1-2p}$ and $\frac{3q}{1-2q}$ are the roots of $lx^2+mx+n=0$ and the greatest common divisor of $l, m, n$ is $1$,then $l-m+n=$

$\alpha$ is the maximum value of $1-2x-5x^2$ and $\beta$ is the minimum value of $x^2-2x+r$. If $5\alpha x^2+\beta x+6>0$ for all real values of $x$,then the interval in which $r$ lies is

Let $x, y, z$ be non-zero real numbers such that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}=7$ and $\frac{y}{x}+\frac{z}{y}+\frac{x}{z}=9$. Then,the value of $\frac{x^3}{y^3}+\frac{y^3}{z^3}+\frac{z^3}{x^3}-3$ is equal to

If $\alpha$ and $\beta$ are the distinct roots of the equation $x^{2}+(3)^{1/4}x+3^{1/2}=0$,then the value of $\alpha^{96}(\alpha^{12}-1) + \beta^{96}(\beta^{12}-1)$ is equal to: