If $a(p + q)^2 + 2bpq + c = 0$ and $a(p + r)^2 + 2bpr + c = 0$,then $qr$ =

If $\alpha, \beta$ are the roots of $ax^2 + bx + c = 0$ and $\alpha + \beta, \alpha^2 + \beta^2, \alpha^3 + \beta^3$ are in $G.P.$, where $\Delta = b^2 - 4ac$, then:

If the sum of the roots of the quadratic equation $ax^2 + bx + c = 0, (abc \neq 0)$ is equal to the sum of the squares of their reciprocals,then $a/c, b/a, c/b$ are in

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 $\alpha, \beta$ are the roots of the quadratic equation $x^{2}+p x+q=0,$ then the values of $\alpha^{3}+\beta^{3}$ and $\alpha^{4}+\alpha^{2} \beta^{2}+\beta^{4}$ are respectively

If $\alpha, \beta$ are the roots of the equation $x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0$,then find the equation whose roots are $\alpha+\frac{1}{\beta}$ and $\beta+\frac{1}{\alpha}$.