Let $$\overrightarrow a $$ and $$\overrightarrow b $$ be two non-zero vectors perpendicular to each other and $$|\overrightarrow a | = |\overrightarrow b |$$. If $$|\overrightarrow a \times \overrightarrow b | = |\overrightarrow a |$$, then the angle between the vectors $$\left( {\overrightarrow a + \overrightarrow b + \left( {\overrightarrow a \times \overrightarrow b } \right)} \right)$$ and $${\overrightarrow a }$$ is equal to :

Let a complex number be w = 1 $$-$$ $${\sqrt 3 }$$i. Let another complex number z be such that |zw| = 1 and arg(z) $$-$$ arg(w) = $${\pi \over 2}$$. Then the area of the triangle with vertices origin, z and w is equal to :

Let f : R $$ \to $$ R be a function defined as

$$f(x) = \left\{ \matrix{ {{\sin (a + 1)x + \sin 2x} \over {2x}},if\,x < 0 \hfill \cr b,\,if\,x\, = 0 \hfill \cr {{\sqrt {x + b{x^3}} - \sqrt x } \over {b{x^{5/2}}}},\,if\,x > 0 \hfill \cr} \right.$$

If f is continuous at x = 0, then the value of a + b is equal to :

Let g(x) = $$\int_0^x {f(t)dt} $$, where f is continuous function in [ 0, 3 ] such that $${1 \over 3}$$ $$ \le $$ f(t) $$ \le $$ 1 for all t$$\in$$ [0, 1] and 0 $$ \le $$ f(t) $$ \le $$ $${1 \over 2}$$ for all t$$\in$$ (1, 3]. The largest possible interval in which g(3) lies is :