The set of all possible values of $$\theta $$ in the interval
(0, $$\pi $$) for which the points (1, 2) and (sin $$\theta $$, cos $$\theta $$) lie
on the same side of the line x + y = 1 is :

Let A = {X = (x, y, z)^T: PX = 0 and

x² + y² + z² = 1} where

$$P = \left[ {\matrix{ 1 & 2 & 1 \cr { - 2} & 3 & { - 4} \cr 1 & 9 & { - 1} \cr } } \right]$$,

then the set A :

$$\mathop {\lim }\limits_{x \to 0} {\left( {\tan \left( {{\pi \over 4} + x} \right)} \right)^{{1 \over x}}}$$ is equal to :

Let f : (–1, $$\infty $$) $$ \to $$ R be defined by f(0) = 1 and
f(x) = $${1 \over x}{\log _e}\left( {1 + x} \right)$$, x $$ \ne $$ 0. Then the function f :