Find the value of $a$ for which the equations $x^3+ax+1=0$ and $x^4+ax^2+1=0$ have a common root.

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 $a_1, a_2, b_1, b_2, c_1, c_2 \in R$.

$A$ value of $b$ for which the equations $x^2+bx-1=0$ and $x^2+x+b=0$ have one root in common is

The equations $x^2-ax+b=0$ and $x^2+bx-a=0$ have a common root. Then:

If the equations $x^2 + px + q = 0$ and $x^2 + qx + p = 0$ have a common root,then $p + q + 1 = $

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