Let $$\overrightarrow x ,\overrightarrow y $$ and $$\overrightarrow z $$ be three vectors each of magnitude $$\sqrt 2 $$ and the angle between each pair of them is $${\pi \over 3}$$. If $$\overrightarrow a $$ is a non-zero vector perpendicular to $$\overrightarrow x $$ and $$\overrightarrow y \times \overrightarrow z $$ and $$\overrightarrow b $$ is a non-zero vector perpendicular to $$\overrightarrow y $$ and $$\overrightarrow z \times \overrightarrow x ,$$ then

Let $$\overrightarrow a \,\,,\,\,\overrightarrow b $$ and $$\overrightarrow c $$ be three non-coplanar unit vectors such that the angle between every pair of them is $${\pi \over 3}.$$ If $$\overrightarrow a \times \overrightarrow b + \overrightarrow b \times \overrightarrow c = p\overrightarrow a + q\overrightarrow b + r\overrightarrow c ,$$ where $$p,q$$ and $$r$$ are scalars, then the value of $${{{p^2} + 2{q^2} + {r^2}} \over {{q^2}}}$$ is

Let $$f:(a,b) \to [1,\infty )$$ be a continuous function and g : R $$\to$$ R be defined as $$g(x) = \left\{ {\matrix{ 0 & , & {x < a} \cr {\int_a^x {f(t)dt} } & , & {a \le x \le b} \cr {\int_a^b {f(t)dt} } & , & {x > b} \cr } } \right.$$ Then,

For every pair of continuous function f, g : [0, 1] $$\to$$ R such that max {f(x) : x $$\in$$ [0, 1]} = max {g(x) : x $$\in$$ [0, 1]}. The correct statement(s) is (are)