$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ are nonzero vectors such that $\overline{\mathrm{a}}$ is perpendicular to $\overline{\mathrm{b}}$ and $\overline{\mathrm{c}},|\overline{\mathrm{a}}|=1,|\overline{\mathrm{~b}}|=2,|\overline{\mathrm{c}}|=1$ and $\overline{\mathrm{b}} \cdot \overline{\mathrm{c}}=1$. There is nonzero vector $\overline{\mathrm{d}}$ coplanar with $\overline{\mathrm{a}}+\overline{\mathrm{b}}$ and $2 \overline{\mathrm{~b}}-\overline{\mathrm{c}}$. If $\overline{\mathrm{d}} \cdot \overline{\mathrm{a}}=1$, then $|\overline{\mathrm{d}}|^2=$

If $\sin ^{-1} x+\sin ^{-1} y=\frac{\pi}{3}$ and $\cot ^{-1}\left(\frac{1}{x}\right)-\cot ^{-1}\left(\frac{1}{y}\right)=0$ then $2 x^2+y^2-x y=$ $\qquad$

The value of $\sin \left[\tan ^{-1}\left(\frac{1-x^2}{2 x}\right)+\cos ^{-1}\left(\frac{1-x^2}{1+x^2}\right)\right]$ is

The area of the region bounded by the curve $y=|x-2|$ between $x=1, x=3$ and X -axis is ……