MHTCET 25 Apr 2025 Shift 2 Paper

Concept:Two lines intersect if they are coplanar and not parallel, i.e. the shortest distance between them is zero.Explanation:For the first line, direction vector is $\bar{b}_{1}=3\hat{i}-\hat{j}$.For the second line, direction vector is $\bar{b}_{2}=2\hat{i}+3\hat{k}$.Also, $\bar{a}_{1}=\hat{i}+\hat{j}-\hat{k}$ and $\bar{a}_{2}=4\hat{i}-\hat{k}$.Compute the cross product:$$\bar{b}_{1}\times \bar{b}_{2} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -1 & 0 \\ 2 & 0 & 3 \end{vmatrix} = -3\hat{i}-9\hat{j}+2\hat{k}$$Now compute $\bar{a}_{2}-\bar{a}_{1}$:$$\bar{a}_{2}-\bar{a}_{1} = 3\hat{i}-\hat{j}$$Check coplanarity:$$(\bar{a}_{2}-\bar{a}_{1})\cdot(\bar{b}_{1}\times \bar{b}_{2}) = (3)(-3)+(-1)(-9)+(0)(2)=0$$Since this dot product is zero, the lines are coplanar.Also, $\bar{b}_{1}$ and $\bar{b}_{2}$ are not proportional, so they are not parallel.Therefore, the lines intersect.Check if they are perpendicular:$$\bar{b}_{1}\cdot\bar{b}_{2}=(3)(2)+(-1)(0)+(0)(3)=6\neq 0$$Hence, the lines are not perpendicular.Answer:The lines are intersecting but not perpendicular.Correct option: A. intersecting but not perpendicular.