Let $${n \ge 2}$$ be an integer. Take n distinct points on a circle and join each pair of points by a line segment. Colour the line segment joining every pair of adjacent points by blue and the rest by red. If the number of red and blue line segments are equal, then the value of n is

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

From a point $$P\left( {\lambda ,\lambda ,\lambda } \right),$$ perpendicular $$PQ$$ and $$PR$$ are drawn respectively on the lines $$y=x, z=1$$ and $$y=-x, z=-1.$$ If $$P$$ is such that $$\angle QPR$$ is a right angle, then the possible value(s) of $$\lambda $$ is/(are)

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