If the real part of the complex number $$z = {{3 + 2i\cos \theta } \over {1 - 3i\cos \theta }},\theta \in \left( {0,{\pi \over 2}} \right)$$ is zero, then the value of sin²3$$\theta$$ + cos²$$\theta$$ 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 _____________.