Let E be an ellipse whose axes are parallel to the co-ordinates axes, having its center at (3, $$-$$4), one focus at (4, $$-$$4) and one vertex at (5, $$-$$4). If mx $$-$$ y = 4, m > 0 is a tangent to the ellipse E, then the value of 5m² is equal to _____________.

If $$\int_0^\pi {({{\sin }^3}x){e^{ - {{\sin }^2}x}}dx = \alpha - {\beta \over e}\int_0^1 {\sqrt t {e^t}dt} } $$, then $$\alpha$$ + $$\beta$$ is equal to ____________.

The number of real roots of the equation e^4x $$-$$ e^3x $$-$$ 4e^2x $$-$$ e^x + 1 = 0 is equal to ______________.

Let y = y(x) be the solution of the differential equation dy = e^{$$\alpha$$x + y} dx; $$\alpha$$ $$\in$$ N. If y(log_e2) = log_e2 and y(0) = log_e$$\left( {{1 \over 2}} \right)$$, then the value of $$\alpha$$ is equal to _____________.