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 $k =$

Let $a, b, c, p, q$ be real numbers. Suppose $\alpha, \beta$ are the roots of the equation $x^2+2px+q=0$ and $\alpha, \frac{1}{\beta}$ are the roots of the equation $ax^2+2bx+c=0$,where $\beta^2 \notin \{-1, 0, 1\}$.
$STATEMENT-1$: $(p^2-q)(b^2-ac) \geq 0$ and
$STATEMENT-2$: $b \neq pa$ or $c \neq qa$.

If $f(x)=x^2+ax+2=0$ and $g(x)=x^2+2x+a=0$ have only one real common root,then the sum of the roots of $f(x)+g(x)=0$ is

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

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 $a, b, c$ 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