Let $$y = y(x)$$ be the solution of the differential equation $$(1 - {x^2})dy = \left( {xy + ({x^3} + 2)\sqrt {1 - {x^2}} } \right)dx, - 1 < x < 1$$, and $$y(0) = 0$$. If $$\int_{{{ - 1} \over 2}}^{{1 \over 2}} {\sqrt {1 - {x^2}} y(x)dx = k} $$, then k^$$-$$1 is equal to _____________.

Let S = {E₁, E₂, ........., E₈} be a sample space of a random experiment such that $$P({E_n}) = {n \over {36}}$$ for every n = 1, 2, ........, 8. Then the number of elements in the set $$\left\{ {A \subseteq S:P(A) \ge {4 \over 5}} \right\}$$ is ___________.

The SI unit of a physical quantity is pascal-second. The dimensional formula of this quantity will be :

The distance of the Sun from earth is 1.5 $$\times$$ 10¹¹ m and its angular diameter is (2000) s when observed from the earth. The diameter of the Sun will be :