Let $$\theta \in \left( {0,{\pi \over 2}} \right)$$. If the system of linear equations

$$(1 + {\cos ^2}\theta )x + {\sin ^2}\theta y + 4\sin 3\,\theta z = 0$$

$${\cos ^2}\theta x + (1 + {\sin ^2}\theta )y + 4\sin 3\,\theta z = 0$$

$${\cos ^2}\theta x + {\sin ^2}\theta y + (1 + 4\sin 3\,\theta )z = 0$$

has a non-trivial solution, then the value of $$\theta$$ is :

Let $$f(x) = \cos \left( {2{{\tan }^{ - 1}}\sin \left( {{{\cot }^{ - 1}}\sqrt {{{1 - x} \over x}} } \right)} \right)$$, 0 < x < 1. Then :

The sum of the series

$${1 \over {x + 1}} + {2 \over {{x^2} + 1}} + {{{2^2}} \over {{x^4} + 1}} + ...... + {{{2^{100}}} \over {{x^{{2^{100}}}} + 1}}$$ when x = 2 is :

If $${{}^{20}{C_r}}$$ is the co-efficient of x^r in the expansion of (1 + x)²⁰, then the value of $$\sum\limits_{r = 0}^{20} {{r^2}.{}^{20}{C_r}} $$ is equal to :