Let $$\left( {{x_0},{y_0}} \right)$$ be the solution of the following equations
$$\matrix{ {{{\left( {2x} \right)}^{\ell n2}}\, = {{\left( {3y} \right)}^{\ell n3}}} \cr {{3^{\ell nx}}\, = {2^{\ell ny}}} \cr } $$
Then $${x_0}$$ is

A value of $$b$$ for which the equations $$$\matrix{ {{x^2} + bx - 1 = 0} \cr {{x^2} + x + b = 0} \cr } $$$

have one root in common is

Let $$p$$ and $$q$$ be real numbers such that $$p \ne 0,\,{p^3} \ne q$$ and $${p^3} \ne - q.$$ If $${p^3} \ne - q.$$ and $$\,\beta $$ are nonzero complex numbers satisfying $$\alpha \, + \beta = - p\,$$ and $${\alpha ^3} + {\beta ^3} = q,$$ then a quadratic equation having $${\alpha \over \beta }$$ and $${\beta \over \alpha }$$ as its roots is

Let $$a,\,b,c$$, $$p,q$$ be real numbers. Suppose $$\alpha ,\,\beta $$ are the roots of the equation $${x^2} + 2px + q = 0$$ and $$\alpha ,{1 \over \beta }$$ are the roots of the equation $$a{x^2} + 2bx + c = 0,$$ where $${\beta ^2} \in \left\{ { - 1,\,0,\,1} \right\}$$

STATEMENT - 1 : $$\left( {{p^2} - q} \right)\left( {{b^2} - ac} \right) \ge 0$$

and STATEMENT - 2 : $$b \ne pa$$ or $$c \ne qa$$