The desired product $$X$$ can be prepared by reacting the major product of the reactions in LIST-I with one or more appropriate reagents in LIST-II (given, order of migratory aptitude: aryl > alkyl > hydrogen)


The correct option is

LIST-I contains reactions and LIST-II contains major products.



Match the reaction in LIST-I with one or more products in LIST-II and choose the correct option.

For the given compound $$X,$$ the total number of optically active stereoisomers is ____________.

For any positive integer n, define

$${f_n}:(0,\infty ) \to R$$ as

$${f_n} = \sum\limits_{j = 1}^n {{{\tan }^{ - 1}}} \left( {{1 \over {1 + (x + j)(x + j - 1)}}} \right)$$

for all x$$ \in $$(0, $$\infty $$). (Here, the inverse trigonometric function tan^$$-$$1 x assumes values in $$\left( { - {\pi \over 2},{\pi \over 2}} \right)$$). Then, which of the following statement(s) is (are) TRUE?