Let the maximum area of the triangle that can be inscribed in the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over 4} = 1,\,a > 2$$, having one of its vertices at one end of the major axis of the ellipse and one of its sides parallel to the y-axis, be $$6\sqrt 3 $$. Then the eccentricity of the ellipse is :

Let the area of the triangle with vertices A(1, $$\alpha$$), B($$\alpha$$, 0) and C(0, $$\alpha$$) be 4 sq. units. If the points ($$\alpha$$, $$-$$$$\alpha$$), ($$-$$$$\alpha$$, $$\alpha$$) and ($$\alpha$$², $$\beta$$) are collinear, then $$\beta$$ is equal to :

The number of distinct real roots of the equation

x⁷ $$-$$ 7x $$-$$ 2 = 0 is

A random variable X has the following probability distribution :

The value of P(1 < X < 4 | X $$\le$$ 2) is equal to :