Let the area enclosed by the x-axis, and the tangent and normal drawn to the curve $$4{x^3} - 3x{y^2} + 6{x^2} - 5xy - 8{y^2} + 9x + 14 = 0$$ at the point ($$-$$2, 3) be A. Then 8A is equal to ______________.

Let $$x = \sin (2{\tan ^{ - 1}}\alpha )$$ and $$y = \sin \left( {{1 \over 2}{{\tan }^{ - 1}}{4 \over 3}} \right)$$. If $$S = \{ a \in R:{y^2} = 1 - x\} $$, then $$\sum\limits_{\alpha \in S}^{} {16{\alpha ^3}} $$ is equal to _______________.

The electric current in a circular coil of 2 turns produces a magnetic induction B₁ at its centre. The coil is unwound and in rewound into a circular coil of 5 tuns and the same current produces a magnetic induction B₂ at its centre. The ratio of $${{{B_2}} \over {{B_1}}}$$ is

A drop of liquid of density $$\rho$$ is floating half immersed in a liquid of density $${\sigma}$$ and surface tension $$7.5 \times 10^{-4}$$ Ncm^$$-$$1. The radius of drop in $$\mathrm{cm}$$ will be :

(g = 10 ms^$$-$$2)