Mathematics Question from JEE Main 2023 · 6 April, Shift 1

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