If the planes \(\vec{r}\cdot(2\hat{\imath} - \lambda\hat{\jmath} + 3\hat{k}) = 0\) and \(\vec{r}\cdot(\lambda\hat{\imath} + 5\hat{\jmath} - \hat{k}) = 5\) are perpendicular to each other, then value of \(\lambda^2 + \lambda\) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 0
Two planes are perpendicular when their normal vectors are perpendicular, that is, their dot product is 0. The normals are
\(\vec{n}_1 = 2\hat{\imath} - \lambda\hat{\jmath} + 3\hat{k}\)
\(\vec{n}_2 = \lambda\hat{\imath} + 5\hat{\jmath} - \hat{k}\)
So
\(\vec{n}_1\cdot\vec{n}_2 = 2\lambda - 5\lambda - 3 = 0\)
\(-3\lambda = 3\)
\(\lambda = -1\)
Hence
\(\lambda^2 + \lambda = 1 - 1 = 0\)
Note: This topic was not in the CUET 2026 syllabus.