Let the tangent to the circle C₁ : x² + y² = 2 at the point M($$-$$1, 1) intersect the circle C₂ : (x $$-$$ 3)² + (y $$-$$ 2)² = 5, at two distinct points A and B. If the tangents to C₂ at the points A and B intersect at N, then the area of the triangle ANB is equal to :

Let the mean and the variance of 5 observations x₁, x₂, x₃, x₄, x₅ be $${24 \over 5}$$ and $${194 \over 25}$$ respectively. If the mean and variance of the first 4 observation are $${7 \over 2}$$ and a respectively, then (4a + x₅) is equal to:

Let $$S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\} $$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at z₁ $$\in$$ S and z₂ $$\in$$ S. If $$5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5 $$, where $$\alpha$$ and $$\beta$$ are integers, then the value of $$\alpha$$ + $$\beta$$ is equal to ___________.

Let y = y(x) be the solution of the differential equation $${{dy} \over {dx}} + {{\sqrt 2 y} \over {2{{\cos }^4}x - {{\cos }^2}x}} = x{e^{{{\tan }^{ - 1}}(\sqrt 2 \cot 2x)}},\,0 < x < {\pi \over 2}$$ with $$y\left( {{\pi \over 4}} \right) = {{{\pi ^2}} \over {32}}$$. If $$y\left( {{\pi \over 3}} \right) = {{{\pi ^2}} \over {18}}{e^{ - {{\tan }^{ - 1}}(\alpha )}}$$, then the value of 3$$\alpha$$² is equal to ___________.