For what value of $a$ do the equations $x^2 - 3x + a = 0$ and $x^2 + ax - 3 = 0$ have a common root?

If $x^2 - hx - 21 = 0$ and $x^2 - 3hx + 35 = 0$ $(h > 0)$ have a common root,then the value of $h$ is equal to

If the equations $x^2+px+2=0$ and $x^2+x+2p=0$ have a common root,then the sum of the roots of the equation $x^2+2px+8=0$ 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 $a_1, a_2, b_1, b_2, c_1, c_2 \in R$.

The quadratic equations $x^2 - 6x + a = 0$ and $x^2 - cx + 6 = 0$ have a common root. The other roots of the first and second equations are integers in the ratio $4:3$. Find the common root.

If the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ $(a \neq b)$ have a common root,then $a+b$ is equal to