The major product of the following reaction is :

The reagent(s) required for the following conversion are :

Consider the following two binary relations on the set A = {a, b, c} :
R₁ = {(c, a), (b, b), (a, c), (c, c), (b, c), (a, a)} and
R₂ = {(a, b), (b, a), (c, c), (c, a), (a, a), (b, b), (a, c)}.
Then :

If $$\lambda $$ $$ \in $$ R is such that the sum of the cubes of the roots of the equation,
x² + (2 $$-$$ $$\lambda $$) x + (10 $$-$$ $$\lambda $$) = 0 is minimum, then the magnitude of the difference of the roots of this equation is :