If $\alpha$ and $\beta$ are the roots of the equation $x^2 + 6x + \lambda = 0$ and $3\alpha + 2\beta = -20$,then $\lambda = $

If $c$ and $d$ are the roots of $x^2+ax+b=0$,then a root of $x^2+(4c+a)x+(b+2ac+4c^2)=0$ is

If $\alpha$ and $\beta$ are the roots of $6x^2 - 6x + 1 = 0$,then the value of $\frac{1}{2}[a + b\alpha + c\alpha^2 + d\alpha^3] + \frac{1}{2}[a + b\beta + c\beta^2 + d\beta^3]$ is

If ${x^2} + px + 1$ is a factor of the expression $a{x^3} + bx + c$,then

Let $\alpha, \beta$ be the roots of the equation $x^{2}-6x-2=0$ with $\alpha>\beta$. If $a_{n}=\alpha^{n}-\beta^{n}$ for $n \geq 1$, then the value of $\frac{a_{10}-2a_{8}}{2a_{9}}$ is

Let $\alpha, \beta$ be the roots of the equation $x^2-px+r=0$ and $\frac{\alpha}{2}, 2\beta$ be the roots of the equation $x^2-qx+r=0$. Then the value of $r$ is