The equations of two sides $$\mathrm{AB}$$ and $$\mathrm{AC}$$ of a triangle $$\mathrm{ABC}$$ are $$4 x+y=14$$ and $$3 x-2 y=5$$, respectively. The point $$\left(2,-\frac{4}{3}\right)$$ divides the third side $$\mathrm{BC}$$ internally in the ratio $$2: 1$$, the equation of the side $$\mathrm{BC}$$ is

If $$\sin x=-\frac{3}{5}$$, where $$\pi< x <\frac{3 \pi}{2}$$, then $$80\left(\tan ^2 x-\cos x\right)$$ is equal to

Let $$[t]$$ be the greatest integer less than or equal to $$t$$. Let $$A$$ be the set of all prime factors of 2310 and $$f: A \rightarrow \mathbb{Z}$$ be the function $$f(x)=\left[\log _2\left(x^2+\left[\frac{x^3}{5}\right]\right)\right]$$. The number of one-to-one functions from $$A$$ to the range of $$f$$ is

If the set $$R=\{(a, b): a+5 b=42, a, b \in \mathbb{N}\}$$ has $$m$$ elements and $$\sum_\limits{n=1}^m\left(1-i^{n !}\right)=x+i y$$, where $$i=\sqrt{-1}$$, then the value of $$m+x+y$$ is