If the locus of the point, whose distances from the point $$(2,1)$$ and $$(1,3)$$ are in the ratio $$5: 4$$, is $$a x^2+b y^2+c x y+d x+e y+170=0$$, then the value of $$a^2+2 b+3 c+4 d+e$$ is equal to :

Let $$[t]$$ denote the greatest integer less than or equal to $$t$$. Let $$f:[0, \infty) \rightarrow \mathbf{R}$$ be a function defined by $$f(x)=\left[\frac{x}{2}+3\right]-[\sqrt{x}]$$. Let $$\mathrm{S}$$ be the set of all points in the interval $$[0,8]$$ at which $$f$$ is not continuous. Then $$\sum_\limits{\text {aes }} a$$ is equal to __________.

In a triangle $$\mathrm{ABC}, \mathrm{BC}=7, \mathrm{AC}=8, \mathrm{AB}=\alpha \in \mathrm{N}$$ and $$\cos \mathrm{A}=\frac{2}{3}$$. If $$49 \cos (3 \mathrm{C})+42=\frac{\mathrm{m}}{\mathrm{n}}$$, where $$\operatorname{gcd}(m, n)=1$$, then $$m+n$$ is equal to _________.

Let $$[t]$$ denote the largest integer less than or equal to $$t$$. If $$\int_\limits0^3\left(\left[x^2\right]+\left[\frac{x^2}{2}\right]\right) \mathrm{d} x=\mathrm{a}+\mathrm{b} \sqrt{2}-\sqrt{3}-\sqrt{5}+\mathrm{c} \sqrt{6}-\sqrt{7}$$, where $$\mathrm{a}, \mathrm{b}, \mathrm{c} \in \mathbf{Z}$$, then $$\mathrm{a}+\mathrm{b}+\mathrm{c}$$ is equal to __________.