If $$\mathop {\lim }\limits_{x \to 0} {[1 + x\ln (1 + {b^2})]^{1/x}} = 2b{\sin ^2}\theta $$, $$b > 0$$ and $$\theta \in ( - \pi ,\pi ]$$, then the value of $$\theta$$ is

Let f(x) = x² and g(x) = sin x for all x $$\in$$ R. Then the set of all x satisfying $$(f \circ g \circ g \circ f)(x) = (g \circ g \circ f)(x)$$, where $$(f \circ g)(x) = f(g(x))$$, is

Let $$\omega$$ $$\ne$$ 1 be a cube root of unity and S be the set of all non-singular matrices of the form $$\left[ {\matrix{ 1 & a & b \cr \omega & 1 & c \cr {{\omega ^2}} & \omega & 1 \cr } } \right]$$, where each of a, b, and c is either $$\omega$$ or $$\omega$$². Then the number of distinct matrices in the set S is

If $$f(x) = \left\{ {\matrix{ { - x - {\pi \over 2},} & {x \le - {\pi \over 2}} \cr { - \cos x} & { - {\pi \over 2} < x \le 0} \cr {x - 1} & {0 < x \le 1} \cr {\ln x} & {x > 1} \cr } } \right.$$, then