The point $$P\left( { - 2\sqrt 6 ,\sqrt 3 } \right)$$ lies on the hyperbola $${{{x^2}} \over {{a^2}}} - {{{y^2}} \over {{b^2}}} = 1$$ having eccentricity $${{\sqrt 5 } \over 2}$$. If the tangent and normal at P to the hyperbola intersect its conjugate axis at the point Q and R respectively, then QR is equal to :

Let y(x) be the solution of the differential equation

2x² dy + (e^y $$-$$ 2x)dx = 0, x > 0. If y(e) = 1, then y(1) is equal to :

Consider the two statements :

(S1) : (p $$\to$$ q) $$ \vee $$ ($$ \sim $$ q $$\to$$ p) is a tautology .

(S2) : (p $$ \wedge $$ $$ \sim $$ q) $$ \wedge $$ ($$\sim$$ p $$\wedge$$ q) is a fallacy.

Then :

The domain of the function $${{\mathop{\rm cosec}\nolimits} ^{ - 1}}\left( {{{1 + x} \over x}} \right)$$ is :