Let y = y(x) be a solution of the differential equation,

$$\sqrt {1 - {x^2}} {{dy} \over {dx}} + \sqrt {1 - {y^2}} = 0$$, |x| < 1.

If $$y\left( {{1 \over 2}} \right) = {{\sqrt 3 } \over 2}$$, then $$y\left( { - {1 \over {\sqrt 2 }}} \right)$$ is equal to :

If the equation, x² + bx + 45 = 0 (b $$ \in $$ R) has conjugate complex roots and they satisfy |z +1| = 2$$\sqrt {10} $$ , then :

The graph which depicts the results of Rutherford gold foil experiment with $$\alpha $$-particales is :
$$\theta $$ : Scattering angle
Y : Number of scattered $$\alpha $$-particles detected (Plots are schematic and not to scale)

The critical angle of a medium for a specific wavelength, if the medium has relative permittivity 3 and relative permeability $${4 \over 3}$$ for this wavelength, will be :