1
JEE Main 2023 (Online) 10th April Evening Shift
Numerical
+4
-1
Out of Syllabus

Let the foot of perpendicular from the point $$\mathrm{A}(4,3,1)$$ on the plane $$\mathrm{P}: x-y+2 z+3=0$$ be N. If B$$(5, \alpha, \beta), \alpha, \beta \in \mathbb{Z}$$ is a point on plane P such that the area of the triangle ABN is $$3 \sqrt{2}$$, then $$\alpha^{2}+\beta^{2}+\alpha \beta$$ is equal to ___________.

2
JEE Main 2023 (Online) 8th April Evening Shift
Numerical
+4
-1
Out of Syllabus

Let $$\mathrm{P}_{1}$$ be the plane $$3 x-y-7 z=11$$ and $$\mathrm{P}_{2}$$ be the plane passing through the points $$(2,-1,0),(2,0,-1)$$, and $$(5,1,1)$$. If the foot of the perpendicular drawn from the point $$(7,4,-1)$$ on the line of intersection of the planes $$P_{1}$$ and $$P_{2}$$ is $$(\alpha, \beta, \gamma)$$, then $$\alpha+\beta+\gamma$$ is equal to ___________.

3
JEE Main 2023 (Online) 8th April Morning Shift
Numerical
+4
-1
Out of Syllabus

Let $$\lambda_{1}, \lambda_{2}$$ be the values of $$\lambda$$ for which the points $$\left(\frac{5}{2}, 1, \lambda\right)$$ and $$(-2,0,1)$$ are at equal distance from the plane $$2 x+3 y-6 z+7=0$$. If $$\lambda_{1} > \lambda_{2}$$, then the distance of the point $$\left(\lambda_{1}-\lambda_{2}, \lambda_{2}, \lambda_{1}\right)$$ from the line $$\frac{x-5}{1}=\frac{y-1}{2}=\frac{z+7}{2}$$ is ____________.

4
JEE Main 2023 (Online) 6th April Evening Shift
Numerical
+4
-1

If the lines $$\frac{x-1}{2}=\frac{2-y}{-3}=\frac{z-3}{\alpha}$$ and $$\frac{x-4}{5}=\frac{y-1}{2}=\frac{z}{\beta}$$ intersect, then the magnitude of the minimum value of $$8 \alpha \beta$$ is _____________.