1
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1

The shortest distance between the lines $${{x + 2} \over 1} = {y \over { - 2}} = {{z - 5} \over 2}$$ and $${{x - 4} \over 1} = {{y - 1} \over 2} = {{z + 3} \over 0}$$ is :

A
8
B
7
C
6
D
9
2
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1
Out of Syllabus

Let two vertices of a triangle ABC be (2, 4, 6) and (0, $$-$$2, $$-$$5), and its centroid be (2, 1, $$-$$1). If the image of the third vertex in the plane $$x+2y+4z=11$$ is $$(\alpha,\beta,\gamma)$$, then $$\alpha\beta+\beta\gamma+\gamma\alpha$$ is equal to :

A
72
B
74
C
76
D
70
3
JEE Main 2023 (Online) 10th April Morning Shift
+4
-1
Out of Syllabus

Let P be the point of intersection of the line $${{x + 3} \over 3} = {{y + 2} \over 1} = {{1 - z} \over 2}$$ and the plane $$x+y+z=2$$. If the distance of the point P from the plane $$3x - 4y + 12z = 32$$ is q, then q and 2q are the roots of the equation :

A
$${x^2} + 18x - 72 = 0$$
B
$${x^2} - 18x - 72 = 0$$
C
$${x^2} + 18x + 72 = 0$$
D
$${x^2} - 18x + 72 = 0$$
4
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1
Out of Syllabus

For $$\mathrm{a}, \mathrm{b} \in \mathbb{Z}$$ and $$|\mathrm{a}-\mathrm{b}| \leq 10$$, let the angle between the plane $$\mathrm{P}: \mathrm{ax}+y-\mathrm{z}=\mathrm{b}$$ and the line $$l: x-1=\mathrm{a}-y=z+1$$ be $$\cos ^{-1}\left(\frac{1}{3}\right)$$. If the distance of the point $$(6,-6,4)$$ from the plane P is $$3 \sqrt{6}$$, then $$a^{4}+b^{2}$$ is equal to :

A
48
B
85
C
32
D
25
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