Q. The number of the distinct real values of λ, for which the vectors $- \lambda^{2}\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}$ , $\widehat{\dot{\mathbf{i}}} - \lambda^{2}\widehat{\dot{\mathbf{j}}} + \widehat{\mathbf{k}}\$ and $\widehat{\dot{\mathbf{i}}} + \widehat{\dot{\mathbf{j}}} - \lambda^{2}\widehat{\mathbf{k}}\$ are coplanar, is
A) Zero
B) One
C) Two
D) Three
Explanation
(c)
Given vectors will be coplanar, if $\left| \begin{matrix}
–\lambda^{2} & 1 & 1 \\
1 & –\lambda^{2} & 1 \\
1 & 1 & - \lambda^{2}
\end{matrix} \right| = 0$
$\Longrightarrow \ \lambda^{6} - 3\lambda^{2} - 2 = 0$
$\Longrightarrow (1 + \lambda^{2})^{2}\left( \lambda^{2} - 2 \right) = 0 \Longrightarrow \lambda^{\ } = \pm \sqrt{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.