MHTCET 15 Apr 2026 Shift 1 Paper

Concept:For a line, every statement can be checked by using its fixed direction vector and a known point on the line.Explanation:The given line is $\frac{x+1}{1}=\frac{y-2}{2}=\frac{z+3}{3}$.A point on this line is $(-1, 2, -3)$ and its direction vector is $\vec{d} = \hat{i} + 2\hat{j} + 3\hat{k}$.Check option A:The point $(-2, 0, -6)$ gives $\frac{-2+1}{1} = \frac{0-2}{2} = \frac{-6+3}{3} = -1$.Since it lies on the same line, statement A is correct.Check option B:For the plane $x - 2y + z + 8 = 0$, the normal vector is $\vec{n} = \hat{i} - 2\hat{j} + \hat{k}$.$\vec{d} \cdot \vec{n} = 1(1) + 2(-2) + 3(1) = 1 - 4 + 3 = 0$, so the line is parallel to the plane.Also, $(-1) - 2(2) + (-3) + 8 = 0$, so the point lies on the plane.Hence, the line lies in the plane. Statement B is correct.Check option C:For the plane $x - 2y + z = 0$, the normal vector is $\vec{n} = (1, -2, 1)$.For a line to be perpendicular to a plane, $\vec{d}$ must be parallel to $\vec{n}$.But $\frac{1}{1} \neq \frac{2}{-2}$, so they are not parallel.Thus, the line is not perpendicular to this plane. Statement C is incorrect.Check option D:The point $(0, 4, 0)$ satisfies $\frac{0+1}{1} = \frac{4-2}{2} = \frac{0+3}{3} = 1$.So it lies on the line. Statement D is correct.Answer:The incorrect statement is Option C.