A) only one value of \[\lambda \]
B) no value of \[\lambda \]
C) only three values of\[\lambda \]
D) only two values of\[\lambda \]
Correct Answer: B
Solution :
(\[[\lambda (\overrightarrow{a}+\overrightarrow{b}){{\lambda }^{2}}\overrightarrow{b}\lambda \overrightarrow{c}]=[\overrightarrow{a}\overrightarrow{b}+\overrightarrow{c}\overrightarrow{b}]\] \[\Rightarrow \]\[\left| \begin{matrix} \lambda ({{a}_{1}}+{{b}_{1}}) & \lambda ({{a}_{2}}+{{b}_{2}}) & \lambda ({{a}_{3}}+{{b}_{3}}) \\ {{\lambda }^{2}}{{b}_{1}} & {{\lambda }^{2}}{{b}_{2}} & {{\lambda }^{2}}{{b}_{3}} \\ \lambda {{c}_{1}} & \lambda {{c}_{2}} & \lambda {{c}_{3}} \\ \end{matrix} \right|\] \[=\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}}+{{c}_{1}} & {{b}_{2}}+{{c}_{2}} & {{b}_{3}}+{{c}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ \end{matrix} \right|\] \[\Rightarrow \]\[{{\lambda }^{4}}\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right|=\left| \begin{matrix} {{a}_{1}} & {{a}_{2}} & {{a}_{3}} \\ {{b}_{1}} & {{b}_{2}} & {{b}_{3}} \\ {{c}_{1}} & {{c}_{2}} & {{c}_{3}} \\ \end{matrix} \right|\] \[\Rightarrow \] \[{{\lambda }^{4}}=-1\] \[\therefore \]There is no real value of\[\lambda \].
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