The value of $a$ for which the equations $x^2 - 3x + a = 0$ and $x^2 + ax - 3 = 0$ have a common root is

If $a, b, c$ are in $A$.$P$. and if the equations $(b-c) x^2+(c-a) x+(a-b)=0$ and $2(c+a) x^2+(b+c) x=0$ have a common root, then

If the equations $x^2 + px + 2q = 0$ and $x^2 + qx + 2p = 0$ $(p \neq q)$ have a common root,then the value of $p + q$ is

Two distinct polynomials $f(x)$ and $g(x)$ are defined as follows: $f(x)=x^2+ax+2$ and $g(x)=x^2+2x+a$. If the equations $f(x)=0$ and $g(x)=0$ have a common root,then the sum of the roots of the equation $f(x)+g(x)=0$ is:

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

Let $a, b, c, d \in \mathbb{R}$. If the equations $2bx^2 + 3cx - d = 0$ and $2ax^2 + 3bx + 4c = 0$ have a common root and $\frac{4bc + ad}{k(b^2 - ac)} = \frac{bd + 4c^2}{4bc + ad}$,then $k =$