If the quadratic equations $3x^2 + ax + 1 = 0$ and $2x^2 + bx + 1 = 0$ have a common root,then what is the value of the expression $5ab - 2a^2 - 3b^2$?

For $a \neq b$,if the equations $x^2+ax+b=0$ and $x^2+bx+a=0$ have a common root,then the value of $a+b=$

Let $\alpha$ be a common root of the equations $x^3-2x-25\lambda=0$ and $3x^3-8x-\frac{175}{3}\lambda=0$,where $\lambda > 0$. Then $\lambda=$

Let $p_1(x) = x^3 - 2020x^2 + b_1x + c_1$ and $p_2(x) = x^3 - 2021x^2 + b_2x + c_2$ be polynomials having two common roots $\alpha$ and $\beta$. Suppose there exist polynomials $q_1(x)$ and $q_2(x)$ such that $p_1(x)q_1(x) + p_2(x)q_2(x) = x^2 - 3x + 2$. Then the correct identity is

If both the roots of $k(6x^2 + 3) + rx + 2x^2 - 1 = 0$ and $6k(2x^2 + 1) + px + 4x^2 - 2 = 0$ are common,then $2r - p$ is equal to

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