If $2i + 3j$,$i + j + k$,and $\lambda i + 4j + 2k$ taken in an order are coterminous edges of a parallelepiped of volume $2$ cubic units,then the value of $\lambda$ is:

If $\hat{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\hat{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(2 \hat{a}-\hat{b}) \cdot[(\hat{a} \times \hat{b}) \times(\hat{a}+2 \hat{b})]$ is

The volume of a tetrahedron with vertices $A(5, -1, 1)$, $B(7, -4, p)$, $C(1, -6, 10)$, and $D(-1, -3, 7)$ is $11 \text{ cubic units}$. Find one of the values of $p$.

The maximum volume of a parallelepiped (in cubic units) with vectors $\vec{u} = 2a\hat{i} + \hat{k}$, $\vec{v} = a\hat{j} - a\hat{k}$, and $\vec{w} = 3\hat{i} + a\hat{j}$, where $a \in [0, 1]$, as its coterminous edges is:

For any non-zero vectors $\bar{a}, \bar{b}, \bar{c}$,the value of $\bar{a} \cdot [(\bar{b} \times \bar{c}) \times (\bar{a} + \bar{b} + \bar{c})]$ is

If the vectors $\bar{a}=\hat{\imath}-2 \hat{\jmath}+\hat{k}$,$\bar{b}=2 \hat{\imath}-5 \hat{\jmath}+p \hat{k}$ and $\bar{c}=5 \hat{\imath}-9 \hat{\jmath}+4 \hat{k}$ are coplanar,then the value of $p$ is