Let $${E_1} = \left\{ {x \in R:x \ne 1\,and\,{x \over {x - 1}} > 0} \right\}$$ and


$${E_2} = \left\{ \matrix{ x \in {E_1}:{\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right) \hfill \cr is\,a\,real\,number \hfill \cr} \right\}$$

(Here, the inverse trigonometric function $${\sin ^{ - 1}}$$ x assumes values in $$\left[ { - {\pi \over 2},{\pi \over 2}} \right]$$.).

Let f : E₁ $$ \to $$ R be the function defined by f(x) = $${{{\log }_e}\left( {{x \over {x - 1}}} \right)}$$ and g : E₂ $$ \to $$ R be the function defined by g(x) = $${\sin ^{ - 1}}\left( {{{\log }_e}\left( {{x \over {x - 1}}} \right)} \right)$$.


The correct option is :

Let S = {1, 2, 3, .........., 9}. For k = 1, 2, .........., 5, let N_k be the number of subsets of S, each containing five elements out of which exactly k are odd. Then N₁ + N₂ + N₃ + N₄ + N₅ =

Let f₁ : R $$ \to $$ R, f₂ : [0, $$\infty $$) $$ \to $$ R, f₃ : R $$ \to $$ R, and f₄ : R $$ \to $$ [0, $$\infty $$) be defined by

$${f_1}\left( x \right) = \left\{ {\matrix{ {\left| x \right|} & {if\,x < 0,} \cr {{e^x}} & {if\,x \ge 0;} \cr } } \right.$$

f₂(x) = x² ;

$${f_3}\left( x \right) = \left\{ {\matrix{ {\sin x} & {if\,x < 0,} \cr x & {if\,x \ge 0;} \cr } } \right.$$

and

$${f_4}\left( x \right) = \left\{ {\matrix{ {{f_2}\left( {{f_1}\left( x \right)} \right)} & {if\,x < 0,} \cr {{f_2}\left( {{f_1}\left( x \right)} \right) - 1} & {if\,x \ge 0;} \cr } } \right.$$