Four dice are thrown simultaneously and the numbers shown on these dice are recorded in 2 $$\times$$ 2 matrices. The probability that such formed matrix have all different entries and are non-singular, is :

Let a vector $${\overrightarrow a }$$ be coplanar with vectors $$\overrightarrow b = 2\widehat i + \widehat j + \widehat k$$ and $$\overrightarrow c = \widehat i - \widehat j + \widehat k$$. If $${\overrightarrow a}$$ is perpendicular to $$\overrightarrow d = 3\widehat i + 2\widehat j + 6\widehat k$$, and $$\left| {\overrightarrow a } \right| = \sqrt {10} $$. Then a possible value of $$[\matrix{ {\overrightarrow a } & {\overrightarrow b } & {\overrightarrow c } \cr } ] + [\matrix{ {\overrightarrow a } & {\overrightarrow b } & {\overrightarrow d } \cr } ] + [\matrix{ {\overrightarrow a } & {\overrightarrow c } & {\overrightarrow d } \cr } ]$$ is equal to :

If $$\int\limits_0^{100\pi } {{{{{\sin }^2}x} \over {{e^{\left( {{x \over \pi } - \left[ {{x \over \pi }} \right]} \right)}}}}dx = {{\alpha {\pi ^3}} \over {1 + 4{\pi ^2}}},\alpha \in R} $$ where [x] is the greatest integer less than or equal to x, then the value of $$\alpha$$ is :

Let three vectors $$\overrightarrow a $$, $$\overrightarrow b $$ and $$\overrightarrow c $$ be such that $$\overrightarrow a \times \overrightarrow b = \overrightarrow c $$, $$\overrightarrow b \times \overrightarrow c = \overrightarrow a $$ and $$\left| {\overrightarrow a } \right| = 2$$. Then which one of the following is not true?