The vectors $\lambda \hat{i} + \hat{j}+2\hat{k}, \hat{i} + \lambda \hat{j} - \hat{k}$ and $2\hat{i}-\hat{j} + \lambda \hat{k}$ are coplanar if :

$\lambda = -2, \lambda = 1 ± \sqrt{3}$

The correct answer is Option (1) → $\lambda = -2, \lambda = 1 ± \sqrt{3}$

The given vector are coplanar if their scalar triple product is 0

so $\begin{vmatrix}λ&1&2\\1&λ&-1\\2&-1&λ\end{vmatrix}=0$

$λ(λ^2-1)+1(-2-λ)+2(-1-2λ)=0$

$λ^3-λ-2-λ-2-4λ=0$

$λ^3-6λ-4=0$

$λ=-2,1+\sqrt{3},1-\sqrt{3}$