The distance of the point $(1,3,-7)$ from the plane passing through the point $(1,-1,-1)$ having normal perpendicular to both the lines $\frac{x-1}{1}=\frac{y+2}{-2}=\frac{z-4}{3}$ and $\frac{x-2}{2}=\frac{y+1}{-1}=\frac{z+7}{-1}$ is

Domain of definition of the real valued function $f(x)=\sqrt{\sin ^{-1}(2 x)+\frac{\pi}{6}}$ is

The value of m , such that $\frac{x-4}{1}=\frac{y-2}{1}=\frac{z-m}{2}$ lies in the plane $2 x-4 y+z=7$, is

Suppose that $\bar{p}, \bar{q}$ and $\overline{\mathrm{r}}$ are three non-coplanar vectors in $\mathbb{R}^3$. Let the components of a vector $\overline{\mathrm{s}}$ along $\overline{\mathrm{p}}, \overline{\mathrm{q}}$ and $\overline{\mathrm{r}}$ be 4,3 and 5 respectively. If the components of this vector $\overline{\mathrm{s}}$ along $(-\overline{\mathrm{p}}+\overline{\mathrm{q}}+\overline{\mathrm{r}}),(\overline{\mathrm{p}}-\overline{\mathrm{q}}+\overline{\mathrm{r}})$ and $(-\overline{\mathrm{p}}-\overline{\mathrm{q}}+\overline{\mathrm{r}})$ are $x$, $y$ and $z$ respectively, then the value of $2 x+y+z$ is