$\mathrm{S}_1=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}, \mathrm{S}_2=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^2$ and $\mathrm{S}_3=\sum_\limits{\mathrm{r}=1}^{\mathrm{n}} \mathrm{r}^3$, then the value of $\lim _\limits{n \rightarrow \infty} \frac{s_1\left(1+\frac{s_3}{4}\right)}{s_2^2}$ is

The numerical value of $\tan \left(2 \tan ^{-1}\left(\frac{1}{5}\right)+\frac{\pi}{4}\right)$

Let $\bar{a}, \bar{b}$ and $\bar{c}$ be three vectors having magnitudes 1,1 and 2 respectively. If $\overline{\mathrm{a}} \times(\overline{\mathrm{a}} \times \overline{\mathrm{c}})+\overline{\mathrm{b}}=\overline{0}$, then the acute angle between $\overline{\mathrm{a}}$ and $\overline{\mathrm{c}}$ is

Area (in sq.units) lying in the first quadrant and bounded by the circle $x^2+y^2=4$ and the lines $x=0$ and $x=2$ is