For how many values $a \in \mathbb{C}$,do the equations $x^2-8x+7=0$ and $x^2-2ax+49=0$ have a common root?

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

If the equations $2x^2 + 3x + 5\lambda = 0$ and $x^2 + 2x + 3\lambda = 0$ have a common root,then $\lambda = $

If $ax^2 + bx + c = 0$ and $bx^2 + cx + a = 0$ have a common root and $a, b, c$ are non-zero real numbers,then $\frac{a^3 + b^3 + c^3}{abc} = $

If $\alpha, \beta$ and $\gamma$ are three consecutive terms of a non-constant $G.P.$ such that the equations $\alpha x^2 + 2\beta x + \gamma = 0$ and $x^2 + x - 1 = 0$ have a common root,then $\alpha(\beta + \gamma)$ is equal to

If the equations $x^2 - cx + d = 0$ and $x^2 - ax + b = 0$ have one common root and the second equation has two equal roots,then $2(b + d) = \dots$