If $x^2 - 11x + a$ and $x^2 - 14x + 2a$ have a common factor,then $a = ......$

If $\alpha$ is the common root of the quadratic equations $x^2-5x+4a=0$ and $x^2-2ax-8=0$,where $a \in R$,then the value of $\alpha^4-\alpha^3+68$ is

Statement-$I$: If $a, b, c \in R$ and the equations $ax^2 + bx + c = 0$ and $x^2 + 3x + 4 = 0$ have a common root,then $\frac{a+c}{b} = \frac{4}{3}$.
Statement-$II$: If $a_1x^2 + b_1x + c_1 = 0$ and $a_2x^2 + b_2x + c_2 = 0$ have both roots common,then $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$,where $a_1, a_2, b_1, b_2, c_1, c_2 \in R$.

If $\alpha$ is a common root of $x^2-5x+\lambda=0$ and $x^2-8x-2\lambda=0$ $(\lambda \neq 0)$ and $\beta, \gamma$ are the other roots of them respectively,then $\alpha+\beta+\gamma+\lambda=$

If both the roots of $k(6x^2 + 3) + rx + 2x^2 - 1 = 0$ and $6k(2x^2 + 1) + px + 4x^2 - 2 = 0$ are common,then $2r - p$ is equal to

If $a, b, c$ are in $G.P.$,then the equations $ax^2 + 2bx + c = 0$ and $dx^2 + 2ex + f = 0$ have a common root if $\frac{d}{a}, \frac{e}{b}, \frac{f}{c}$ are in