COMEDK 2025 Morning Solved Paper

Concept:The angle between two lines is $\frac{\pi}{2}$ when their direction vectors are perpendicular, i.e., their dot product is zero.Explanation:Extract the direction vectors from the given parametric equations.For the first line: $\vec{r} = \hat{\imath}+\hat{\jmath}+\hat{k} + p(2\hat{\imath}+\hat{\jmath}+2\hat{k})$, so the direction vector is $\vec{d}_1 = 2\hat{\imath} + \hat{\jmath} + 2\hat{k}$.For the second line: $\vec{r} = (1+q)\hat{\imath} + (1+q\lambda)\hat{\jmath} + (1+q)\hat{k}$ can be rewritten as $\vec{r} = \hat{\imath}+\hat{\jmath}+\hat{k} + q(\hat{\imath} + \lambda\hat{\jmath} + \hat{k})$. Hence the direction vector is $\vec{d}_2 = \hat{\imath} + \lambda\hat{\jmath} + \hat{k}$.Set their dot product equal to zero for perpendicularity.$\vec{d}_1 \cdot \vec{d}_2 = (2)(1) + (1)(\lambda) + (2)(1) = 2 + \lambda + 2 = \lambda + 4$.Equate to zero: $\lambda + 4 = 0$.Solving gives $\lambda = -4$.Answer:$\lambda = -4$, which corresponds to option A.