Question

The number of distinct real values of $\lambda$, for which the vectors $-\lambda^{2} \hat{ i }+\hat{ j }+\hat{ k }, \hat{ i }-\lambda^{2} \hat{ j }+\hat{ k }$ and $\hat{ i }+\hat{ j }-\lambda^{2} \hat{ k }$ are coplanar, is

• A. 0
• B. 1
• C. 2
• D. 3

Answer 1

Answer: (C)

Since, given vectors are coplanar
$\therefore \begin{vmatrix}-\lambda^{2} & 1 & 1 \\1 & -\lambda^{2} & 1 \\1 & 1 & -\lambda^{2}\end{vmatrix}=0 $
$\Rightarrow \lambda^{6}-3 \lambda^{2}-2=0 $
$\Rightarrow\left(1+\lambda^{2}\right)^{2}\left(\lambda^{2}-2\right)=0 $
$\Rightarrow \lambda=\pm \sqrt{2}$