The least positive non-integral solution of the equation $\sin \pi\left(x^2+x\right)=\sin \pi x^2$ is

$S=\cot ^{-1}\left(\frac{1+2 \times 6}{4}\right)+\cot ^{-1}\left(\frac{1+3 \times 8}{10}\right)+\cot ^{-1} \left(\frac{1+4 \times 10}{18}\right) \ldots \ldots \ldots$ upto $\infty$ terms is equal to

The area of the region $\left\{(x, y): x y<8,1 \leq y \leq x^2\right\}$ is

If $\alpha$ is a root of $x^4=1$ with negative principal argument, then the principle argument of $\Delta(A)$, where $\Delta(A)=\left|\begin{array}{ccc}1 & 1 & 1 \\ \alpha^n & \alpha^{n+1} & \alpha^{n+3} \\ \frac{1}{\alpha^{n+1}} & \frac{1}{\alpha^n} & 0\end{array}\right|$