A ray of light through (2, 1) is reflected at a point P on the y-axis and then passes through the point (5, 3). If this reflected ray is the directrix of an ellipse with eccentricity $${1 \over 3}$$ and the distance of the nearer focus from this directrix is $${8 \over {\sqrt {53} }}$$, then the equation of the other directrix can be :

If the coefficients of x⁷ in $${\left( {{x^2} + {1 \over {bx}}} \right)^{11}}$$ and x^$$-$$7 in $${\left( {{x} - {1 \over {bx^2}}} \right)^{11}}$$, b $$\ne$$ 0, are equal, then the value of b is equal to :

If $$\sin \theta + \cos \theta = {1 \over 2}$$, then 16(sin(2$$\theta$$) + cos(4$$\theta$$) + sin(6$$\theta$$)) is equal to :

Let $$A = \left[ {\matrix{ 1 & 2 \cr { - 1} & 4 \cr } } \right]$$. If A^$$-$$1 = $$\alpha$$I + $$\beta$$A, $$\alpha$$, $$\beta$$ $$\in$$ R, I is a 2 $$\times$$ 2 identity matrix then 4($$\alpha$$ $$-$$ $$\beta$$) is equal to :