If the area of the parallelogram whose adjacent sides are $3\overline{\mathrm{i}}-4\overline{\mathrm{j}}-\mathrm{\lambda }\overline{\mathrm{k}},2\overline{\mathrm{j}}-4\overline{\mathrm{k}}$ is $\sqrt{436}$ sq.units $(\mathrm{\lambda }\ge 0)$, then $\mathrm{\lambda }$+2=

$$\begin{gathered} \overline{\mathrm{a}}=3\overline{\mathrm{i}}-4\overline{\mathrm{j}}-\mathrm{\lambda }\overline{\mathrm{k}}, \overline{\mathrm{b}}=2\overline{\mathrm{j}}-4\overline{\mathrm{k}} are adjacent sides of the paralle\log ram \\ \text{ Consider }\overline{\mathrm{a}}\times \overline{\mathrm{b}}=\left|\begin{array}{ccc}\overline{\mathrm{i}}& \overline{\mathrm{j}}& \overline{\mathrm{k}}\\ 3& -4& -\mathrm{\lambda }\\ 0& 2& -4\end{array}\right|=(16+2\mathrm{\lambda })\overline{\mathrm{i}}+12\overline{\mathrm{j}}+6\overline{\mathrm{k}} \end{gathered}$$
Area of parallelogram $$\begin{gathered} |\overline{\mathrm{a}}\times \overline{\mathrm{b}}|=\sqrt{436} \\ \Rightarrow \sqrt{(16+2\mathrm{\lambda }{)}^2+144+36}=\sqrt{436} \\ \Rightarrow (16+2\mathrm{\lambda }{)}^2+180=436 \\ \Rightarrow (16+2\mathrm{\lambda }{)}^2=256 \end{gathered}$$
$$\begin{gathered} \Rightarrow \mathrm{\lambda }=0\text{ or }\Rightarrow \mathrm{\lambda }=-6\therefore \mathrm{\lambda }=0(\because \mathrm{\lambda }\ge 0) \\ \Rightarrow \mathrm{\lambda }+2=2 \end{gathered}$$