MHTCET 22 Apr 2025 Shift 1 Paper

Concept:The angle between two lines is found using their direction vectors. If the dot product of these vectors is zero, the lines are perpendicular.Explanation:For the first line, let $3x = 2y = -z = k$.Then $x = \frac{k}{3}$, $y = \frac{k}{2}$, $z = -k$.So its direction vector is proportional to $\left(\frac{1}{3}, \frac{1}{2}, -1\right)$, which simplifies to $(2, 3, -6)$.For the second line, let $-x = 6y = -4z = m$.Then $x = -m$, $y = \frac{m}{6}$, $z = -\frac{m}{4}$.So its direction vector is proportional to $\left(-1, \frac{1}{6}, -\frac{1}{4}\right)$, which simplifies to $(-12, 2, -3)$.Now take the dot product: $(2, 3, -6) \cdot (-12, 2, -3) = -24 + 6 + 18 = 0$.Since the dot product is zero, the direction vectors are perpendicular.Therefore, the angle between the lines is $\frac{\pi}{2}$.Answer:Option C: $\frac{\pi}{2}$