IIT-JEE 2007 · Paper 1, Mathematics question 51

Coplanarity: the determinant of the three vectors vanishes. The matrix is $J-(\lambda^2+1)I$ with $J$ the all-ones matrix (eigenvalues 3, 0, 0), so
$$\det=\left(2-\lambda^2\right)\left(\lambda^2+1\right)^2.$$

Roots. $\lambda^2+1>0$, so $\lambda^2=2$: $\lambda=\pm\sqrt2$, two values.

(C)