MHTCET 16 Apr 2026 Shift 1 Paper

Concept:Two lines in space are skew if they are neither parallel nor intersecting.Explanation:For the first line, the point is $A(1,-2,3)$ and the direction ratios are $\vec{d}_{1}=(-1,1,-2)$.For the second line, the point is $B(1,-2,-1)$ and the direction ratios are $\vec{d}_{2}=(1,1,-2)$.Check if the lines are parallel by comparing direction ratios.Since $\frac{-1}{1} \neq \frac{1}{1}$, the direction ratios are not proportional, so the lines are not parallel.Now, write the parametric equations to check for intersection.For $L_{1}$: $(x,y,z) = (1-t,\,-2+t,\,3-2t)$.For $L_{2}$: $(x,y,z) = (1+s,\,-2+s,\,-1-2s)$.Equating the $x$-coordinates gives $1-t = 1+s$, so $s = -t$.Equating the $y$-coordinates gives $-2+t = -2+s$, so $s = t$.Combining $s = -t$ and $s = t$ gives $t = 0$ and $s = 0$.At $t = 0$, the $z$-coordinate of $L_{1}$ is $3$.At $s = 0$, the $z$-coordinate of $L_{2}$ is $-1$.The $z$-coordinates are not equal, so the lines do not intersect.Since the lines are neither parallel nor intersecting, they are skew lines.Answer:Option D: skew lines.