If $\alpha, \beta, \gamma$ are the roots of the cubic equation $x^3+p_1 x^2+p_2 x+p_3=0$. Let $S_r=\alpha^r+\beta^r+\gamma^r$. Given $S_1=10, S_2=38$ and $S_3=-1840$,then $p_3=$

If $\alpha$ and $\beta$ are the roots of the quadratic equation $ax^{2}+bx+c=0$ and $3b^{2}=16ac$, then:

If $\alpha, \beta, \gamma$ are the roots of $x^3+4x+1=0$,then the equation whose roots are $\frac{\alpha^2}{\beta+\gamma}, \frac{\beta^2}{\gamma+\alpha}, \frac{\gamma^2}{\alpha+\beta}$ is

Let $a, b \in \mathbb{C}$. Let $\alpha, \beta$ be the roots of the equation $x^2 + ax + b = 0$. If $\beta - \alpha = \sqrt{11}i$ and $\beta^2 - \alpha^2 = 3\sqrt{11}i$, then find the value of $|(\beta^3 - \alpha^3)^2|$.

If $a, b$ and $c$ are the roots of $x^3+qx+r=0$,then $(a-b)^2+(b-c)^2+(c-a)^2=$ (in $q$)

If $A$ is the $A.M.$ of the roots of the equation $x^2 - 2ax + b^2 = 0$ and $G$ is the $G.M.$ of the roots of the equation $x^2 - 2bx + a^2 = 0,$ then