Consider a rectangle ABCD having 5, 7, 6, 9 points in the interior of the line segments AB, CD, BC, DA respectively. Let $$\alpha$$ be the number of triangles having these points from different sides as vertices and $$\beta$$ be the number of quadrilaterals having these points from different sides as vertices. Then ($$\beta$$ $$-$$ $$\alpha$$) is equal to :

Let f : S $$ \to $$ S where S = (0, $$\infty $$) be a twice differentiable function such that f(x + 1) = xf(x). If g : S $$ \to $$ R be defined as g(x) = log_e f(x), then the value of |g''(5) $$-$$ g''(1)| is equal to :

Consider the integral
$$I = \int_0^{10} {{{[x]{e^{[x]}}} \over {{e^{x - 1}}}}dx} $$,
where [x] denotes the greatest integer less than or equal to x. Then the value of I is equal to :

Let C₁ be the curve obtained by the solution of differential equation

$$2xy{{dy} \over {dx}} = {y^2} - {x^2},x > 0$$. Let the curve C₂ be the

solution of $${{2xy} \over {{x^2} - {y^2}}} = {{dy} \over {dx}}$$. If both the curves pass through (1, 1), then the area enclosed by the curves C₁ and C₂ is equal to :