The cubic equation whose roots are the squares of the roots of the equation $12x^3-20x^2+x+3=0$ is

Let $\alpha$ and $\beta$ be the roots of the equation $x^2 - 6x - 2 = 0$. If $a_n = \alpha^n - \beta^n$ for $n \ge 1$,then the value of $\frac{a_{10} - 2a_8}{2a_9}$ is equal to:

If the product of the roots of the equation $x^2 - 3kx + 2e^{2\log k} - 1 = 0$ is $7$,then find the value of $k$.

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 $p$ and $q$ be real numbers such that $p \neq 0$,$p^3 \neq q$,and $p^3 \neq -q$. If $\alpha$ and $\beta$ are nonzero complex numbers satisfying $\alpha+\beta = -p$ and $\alpha^3+\beta^3 = q$,then a quadratic equation having $\frac{\alpha}{\beta}$ and $\frac{\beta}{\alpha}$ as its roots is

If $\alpha, \beta$,where $\alpha < \beta$,are the roots of the equation $\lambda x^{2} - (\lambda + 3)x + 3 = 0$ such that $\frac{1}{\alpha} - \frac{1}{\beta} = \frac{1}{3}$,then the sum of all possible values of $\lambda$ is: