Let $$\mathrm{A}(0,1), \mathrm{B}(1,1)$$ and $$\mathrm{C}(1,0)$$ be the mid-points of the sides of a triangle with incentre at the point $$\mathrm{D}$$. If the focus of the parabola $$y^{2}=4 \mathrm{ax}$$ passing through $$\mathrm{D}$$ is $$(\alpha+\beta \sqrt{2}, 0)$$, where $$\alpha$$ and $$\beta$$ are rational numbers, then $$\frac{\alpha}{\beta^{2}}$$ is equal to :
Let S be the set of all values of $$\theta \in[-\pi, \pi]$$ for which the system of linear equations
$$x+y+\sqrt{3} z=0$$
$$-x+(\tan \theta) y+\sqrt{7} z=0$$
$$x+y+(\tan \theta) z=0$$
has non-trivial solution. Then $$\frac{120}{\pi} \sum_\limits{\theta \in \mathrm{s}} \theta$$ is equal to :
Let $$[t]$$ denote the greatest integer function. If $$\int_\limits{0}^{2.4}\left[x^{2}\right] d x=\alpha+\beta \sqrt{2}+\gamma \sqrt{3}+\delta \sqrt{5}$$, then $$\alpha+\beta+\gamma+\delta$$ is equal to __________.
Let the area enclosed by the lines $$x+y=2, \mathrm{y}=0, x=0$$ and the curve $$f(x)=\min \left\{x^{2}+\frac{3}{4}, 1+[x]\right\}$$ where $$[x]$$ denotes the greatest integer $$\leq x$$, be $$\mathrm{A}$$. Then the value of $$12 \mathrm{~A}$$ is _____________.