Let the position vectors of the points A, B, C and D be 5 i +5 j +2 λ k, i +2 j +3 k,-2 i +λ j +4 k and - i +5 j +6 k. Let the set S={λ ∈ R: the…

Let the position vectors of the points $\displaystyle \mathrm{A}, \mathrm{B}, \mathrm{C}$ and D be $\displaystyle 5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}$ and $\displaystyle -\hat{i}+5 \hat{j}+6 \hat{k}$. Let the set $\displaystyle S=\{\lambda \in \mathbb{R}$ : the points $\displaystyle \mathrm{A}, \mathrm{B}, \mathrm{C}$ and D are coplanar $\displaystyle \}$. Then $\displaystyle \sum_{\lambda \in \mathrm{S}}(\lambda+2)^2$ is equal to