The probability that a relation R from {x, y} to {x, y} is both symmetric and transitive, is equal to :

The number of values of a $$\in$$ N such that the variance of 3, 7, 12, a, 43 $$-$$ a is a natural number is :

Let  $$\overrightarrow a = \widehat i - 2\widehat j + 3\widehat k$$,   $$\overrightarrow b = \widehat i + \widehat j + \widehat k$$   and   $$\overrightarrow c $$   be a vector such that   $$\overrightarrow a + \left( {\overrightarrow b \times \overrightarrow c } \right) = \overrightarrow 0 $$   and   $$\overrightarrow b \,.\,\overrightarrow c = 5$$. Then the value of   $$3\left( {\overrightarrow c \,.\,\overrightarrow a } \right)$$   is equal to _________.

Let y = y(x), x > 1, be the solution of the differential equation $$(x - 1){{dy} \over {dx}} + 2xy = {1 \over {x - 1}}$$, with $$y(2) = {{1 + {e^4}} \over {2{e^4}}}$$. If $$y(3) = {{{e^\alpha } + 1} \over {\beta {e^\alpha }}}$$, then the value of $$\alpha + \beta $$ is equal to _________.