A) Exactly three values of \[\lambda \]
B) Exactly two values of \[\lambda \]
C) Exactly one value of \[\lambda \]
D) No value of \[\lambda \]
Correct Answer: D
Solution :
\[[\lambda (\mathbf{a}+\mathbf{b})\,\,{{\lambda }^{2}}\mathbf{b}\,\,\lambda \mathbf{c}]=[\mathbf{a}\,\ \mathbf{b}\,\,+\mathbf{c}\,\,\mathbf{b}]\] Þ \[\lambda (\mathbf{a}+\mathbf{b}).({{\lambda }^{2}}\mathbf{b}\times \lambda \mathbf{c})\]\[=\mathbf{a}.((\mathbf{b}+\mathbf{c})\times \mathbf{b})\] Þ\[\lambda (\mathbf{a}+\mathbf{b}).{{\lambda }^{3}}(\mathbf{b}\times \mathbf{c})\]\[=\mathbf{a}.(\mathbf{b}\times \mathbf{b}+\mathbf{c}\times \mathbf{b})\] Þ \[{{\lambda }^{4}}[\mathbf{a}.(\mathbf{b}\times \mathbf{c})+\mathbf{b}.(\mathbf{b}\times \mathbf{c})]=\mathbf{a}.(\mathbf{c}\times \mathbf{b})\] Þ \[{{\lambda }^{4}}[\mathbf{a}\ \mathbf{b}\ \mathbf{c}]=-[\mathbf{a}\ \mathbf{b}\ \mathbf{c}]\] Þ \[[\mathbf{a}\ \mathbf{b}\ \mathbf{c}]({{\lambda }^{4}}+1)=0\] Since a, b, c are non-coplanar, so \[[\mathbf{a}\ \mathbf{b}\ \mathbf{c}]\ne 0\] \[\therefore \] \[{{\lambda }^{4}}=-1\]. Hence no real value of \[\lambda \].
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