Let $$u$$ and $$v$$ be units vectors. If $$w$$ is a vector such that $$w + \left( {w \times u} \right) = v,$$ then prove that $$\left| {\left( {u \times v} \right) \cdot w} \right| \le 1/2$$ and that the equality holds if and only if $$u$$ is perpendicular to $$v .$$

$$\int\limits_{\pi /4}^{3\pi /4} {{{dx} \over {1 + \cos x}}} $$ is equal to

If two distinct chords, drawn from the point (p, q) on the circle $${x^2}\, + \,{y^2} = \,px\, + \,qy\,\,(\,where\,pq\, \ne \,0)$$ are bisected by the x - axis, then

$$If\,i = \sqrt { - 1} ,\,\,then\,\,4 + 5{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{334}} + 3{\left( { - {1 \over 2} + {{i\sqrt 3 } \over 2}} \right)^{365}}$$ is equal to