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 $$c \ne qa$$

Let $$\alpha ,\,\beta $$ be the roots of the equation $${x^2} - px + r = 0$$ and $${\alpha \over 2},\,2\beta $$ be the roots of the equation $${x^2} - qx + r = 0$$. Then the value of $$r$$

Let $$a,\,b,\,c$$ be the sides of triangle where $$a \ne b \ne c$$ and $$\lambda \in R$$.
If the roots of the equation $${x^2} + 2\left( {a + b + c} \right)x + 3\lambda \left( {ab + bc + ca} \right) = 0$$ are real, then

For all $$'x',{x^2} + 2ax + 10 - 3a > 0,$$ then the interval in which '$$a$$' lies is