$\mathrm{If} \mathrm{AB}\perp \mathrm{BC}, \mathrm{Where} \mathrm{A}(2\mathrm{k}, 2, 3), \mathrm{B}(\mathrm{k}, 1, 5), \mathrm{C}(3+\mathrm{k},2,1)$ then the value of k =

$$\begin{gathered} \mathrm{A}(2\mathrm{k}, 2, 3), \mathrm{B}(\mathrm{k}, 1, 5), \mathrm{C}(3+\mathrm{k},2,1) \\ \mathrm{D}.\mathrm{r}\text{'}\mathrm{s} \mathrm{of}\overrightarrow{\mathrm{AB}}=\left(-\mathrm{k},-1,2\right)=\left({\mathrm{a}}_1,{\mathrm{b}}_1,{\mathrm{c}}_1\right) \mathrm{D}.\mathrm{r}\text{'}\mathrm{s} \overrightarrow{\mathrm{BC}}=(3,1,-4)=({\mathrm{a}}_2,{\mathrm{b}}_2,{\mathrm{c}}_2) \\ \mathrm{Given} \mathrm{AB}\perp \mathrm{BC}, \Rightarrow {\mathrm{a}}_1{\mathrm{a}}_2+{\mathrm{b}}_1{\mathrm{b}}_2+{\mathrm{c}}_1{\mathrm{c}}_2=0\Rightarrow -3\mathrm{k}-1-8=0\Rightarrow \mathrm{k}=-3 \end{gathered}$$