If the solution curve $$y = y(x)$$ of the differential equation $${y^2}dx + ({x^2} - xy + {y^2})dy = 0$$, which passes through the point (1, 1) and intersects the line $$y = \sqrt 3 x$$ at the point $$(\alpha ,\sqrt 3 \alpha )$$, then value of $${\log _e}(\sqrt 3 \alpha )$$ is equal to :

Let $$x = 2t$$, $$y = {{{t^2}} \over 3}$$ be a conic. Let S be the focus and B be the point on the axis of the conic such that $$SA \bot BA$$, where A is any point on the conic. If k is the ordinate of the centroid of the $$\Delta$$SAB, then $$\mathop {\lim }\limits_{t \to 1} k$$ is equal to :

Let a circle C in complex plane pass through the points $${z_1} = 3 + 4i$$, $${z_2} = 4 + 3i$$ and $${z_3} = 5i$$. If $$z( \ne {z_1})$$ is a point on C such that the line through z and z₁ is perpendicular to the line through z₂ and z₃, then $$arg(z)$$ is equal to :

Let C_r denote the binomial coefficient of x^r in the expansion of $${(1 + x)^{10}}$$. If for $$\alpha$$, $$\beta$$ $$\in$$ R, $${C_1} + 3.2{C_2} + 5.3{C_3} + $$ ....... upto 10 terms $$ = {{\alpha \times {2^{11}}} \over {{2^\beta } - 1}}\left( {{C_0} + {{{C_1}} \over 2} + {{{C_2}} \over 3} + \,\,.....\,\,upto\,10\,terms} \right)$$ then the value of $$\alpha$$ + $$\beta$$ is equal to ___________.