Let f(x) = sin($$\pi $$ cos x) and g(x) = cos(2$$\pi $$ sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order:

X = {x : f(x) = 0}, Y = {x : f'(x) = 0}

Z = {x : g(x) = 0}, W = {x : g'(x) = 0}

List - I contains the sets X, Y, Z and W. List - II contains some information regarding these sets.



Which of the following is the only CORRECT combination?

Let f(x) = sin($$\pi $$ cos x) and g(x) = cos(2$$\pi $$ sin x) be two functions defined for x > 0. Define the following sets whose elements are written in the increasing order :

X = {x : f(x) = 0}, Y = {x : f'(x) = 0}

Z = {x : g(x) = 0}, W = {x : g'(x) = 0}

List - I contains the sets X, Y, Z and W. List - II contains some information regarding these sets.

The value of

$$\sum\limits_{k = 1}^{13} {{1 \over {\sin \left( {{\pi \over 4} + {{\left( {k - 1} \right)\pi } \over 6}} \right)\sin \left( {{\pi \over 4} + {{k\pi } \over 6}} \right)}}} $$ is equal to

Let $$S = \left\{ {x \in \left( { - \pi ,\pi } \right):x \ne 0, \pm {\pi \over 2}} \right\}.$$ The sum of all distinct solutions of the equation $$\sqrt 3 \,\sec x + \cos ec\,x + 2\left( {\tan x - \cot x} \right) = 0$$ in the set S is equal to