If $$\alpha, \beta$$ are the roots of the equation

$$ x^{2}-\left(5+3^{\sqrt{\log _{3} 5}}-5^{\sqrt{\log _{5} 3}}\right)x+3\left(3^{\left(\log _{3} 5\right)^{\frac{1}{3}}}-5^{\left(\log _{5} 3\right)^{\frac{2}{3}}}-1\right)=0 $$,

then the equation, whose roots are $$\alpha+\frac{1}{\beta}$$ and $$\beta+\frac{1}{\alpha}$$, is :

The minimum value of the sum of the squares of the roots of $$x^{2}+(3-a) x+1=2 a$$ is:

Let S be the set of all integral values of $$\alpha$$ for which the sum of squares of two real roots of the quadratic equation $$3{x^2} + (\alpha - 6)x + (\alpha + 3) = 0$$ is minimum. Then S :

Let $$\alpha$$ be a root of the equation 1 + x² + x⁴ = 0. Then, the value of $$\alpha$$¹⁰¹¹ + $$\alpha$$²⁰²² $$-$$ $$\alpha$$³⁰³³ is equal to :