If $$\left|f\left(x_{1}\right)-f\left(x_{2}\right)\right| \leq\left(x_{1}-x_{2}\right)^{2}$$, for all $$x_{1}, x_{2} \in$$ $$\mathbb{R}$$. Find the equation of tangent to the curve $$y=f(x)$$ at the point $$(1,2)$$.

If total number of runs scored in $$n$$ matches is $$\left(\frac{n+1}{4}\right)\left(2^{n+1}-n-2\right)$$ where $$n > 1$$, and the runs scored in the $$k^{\text {th }}$$ match are given by $$k .2^{n+1-k}$$, where $$1 \leq k \leq n$$. Find, $$n$$.

The area of the triangle formed by the intersection of a line parallel to X-axis and passing through $$(h, k)$$ with the lines $$y=x$$ and $$x+y=2$$ is $$4 h^{2}$$. Find the locus of point $$P$$.

Evatuate:

$$\int_\limits{0}^{\pi} e^{|\cos x|}\left[2 \sin \left(\frac{1}{2} \cos x\right)+3 \cos \left(\frac{1}{2} \cos x\right)\right] \sin x ~d x$$