MHTCET 25 Apr 2025 Shift 2 Paper

Concept:If a line lies in a plane, its direction vector is perpendicular to the plane's normal, and every point on the line satisfies the plane equation.Explanation:The direction ratios of the line are $(2,-1,2)$.The direction ratios of the normal to the plane are $(\alpha,3,-1)$.Since the line lies in the plane, the dot product of these direction vectors is zero.$$2\alpha + (-1)(3) + 2(-1) = 0$$$$2\alpha - 3 - 2 = 0$$$$2\alpha = 5 \Rightarrow \alpha = \frac{5}{2}$$The point $(3,-5,-2)$ lies on the line, so it also lies on the plane.Substitute this point into the plane equation:$$\frac{5}{2}(3) + 3(-5) - (-2) + \beta = 0$$$$\frac{15}{2} - 15 + 2 + \beta = 0$$$$\frac{15}{2} - 13 + \beta = 0$$$$\beta = \frac{11}{2}$$Answer:$\alpha = \frac{5}{2}$, $\beta = \frac{11}{2}$; hence the correct option is D.