$\overline{\mathrm{a}}, \overline{\mathrm{b}}, \overline{\mathrm{c}}$ 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 $|\overline{\mathrm{d}}|^2=$

The value of $m \in \mathbb{R}$, when angle between the vectors $\overline{\mathrm{p}}=\mathrm{m} y \hat{\mathrm{i}}-6 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}$ and $\overline{\mathrm{q}}=y \hat{\mathrm{i}}+2 \hat{\mathrm{j}}+2 \mathrm{~m} y \hat{\mathrm{k}}$ is obtuse angle, is

The volume of the tetrahedron whose coterminous edges are represented by

$$ \bar{a}=-12 \hat{i}+p \hat{k}, \bar{b}=3 \hat{j},-\hat{k}, \bar{c}=2 \hat{i}+\hat{j}-15 \hat{k} $$

570 cu. units, then $\mathrm{p}=$

The maximum value and minimum value of the volume of the parallelopiped having coterminous edges $\hat{\mathrm{i}}+x \hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{j}}+x \hat{\mathrm{k}}$ and $x \hat{\mathrm{i}}+\hat{\mathrm{k}}$ are respectively