Given that $$\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = 3$$ and $$\overrightarrow a \times \overrightarrow b = \overrightarrow c ,$$ then $$\overrightarrow b \, = $$.........

Determine the value of $$'c'$$ so that for all real $$x,$$ the vector
$$cx\widehat i - 6\widehat j - 3\widehat k$$ and $$x\widehat i + 2\widehat j + 2cx\widehat k$$ make an obtuse angle with each other.

If $$\exp \,\,\,\left\{ {\left( {\left( {{{\sin }^2}x + {{\sin }^4}x + {{\sin }^6}x + \,\,\,..............\infty } \right)\,In\,\,2} \right)} \right\}$$ satiesfies the equation $${x^2} - 9x + 8 = 0,$$ find the value of $${{\cos x} \over {\cos x + \sin x}},\,0 < x < {\pi \over 2}.$$

If the mean and the variance of binomial variate $$X$$ are $$2$$ and $$1$$ respectively, then the probability that $$X$$ takes a value greater than one is equal to ...............