1
JEE Main 2022 (Online) 28th June Evening Shift
Numerical
+4
-1

Let the image of the point P(1, 2, 3) in the line $$L:{{x - 6} \over 3} = {{y - 1} \over 2} = {{z - 2} \over 3}$$ be Q. Let R ($$\alpha$$, $$\beta$$, $$\gamma$$) be a point that divides internally the line segment PQ in the ratio 1 : 3. Then the value of 22 ($$\alpha$$ + $$\beta$$ + $$\gamma$$) is equal to __________.

2
JEE Main 2022 (Online) 27th June Morning Shift
Numerical
+4
-1
Out of Syllabus

Let the mirror image of the point (a, b, c) with respect to the plane 3x $$-$$ 4y + 12z + 19 = 0 be (a $$-$$ 6, $$\beta$$, $$\gamma$$). If a + b + c = 5, then 7$$\beta$$ $$-$$ 9$$\gamma$$ is equal to ______________.

3
JEE Main 2022 (Online) 25th June Evening Shift
Numerical
+4
-1

Let l1 be the line in xy-plane with x and y intercepts $${1 \over 8}$$ and $${1 \over {4\sqrt 2 }}$$ respectively, and l2 be the line in zx-plane with x and z intercepts $$- {1 \over 8}$$ and $$- {1 \over {6\sqrt 3 }}$$ respectively. If d is the shortest distance between the line l1 and l2, then d$$-$$2 is equal to _______________.

4
JEE Main 2022 (Online) 25th June Morning Shift
Numerical
+4
-1
Out of Syllabus

Let the lines

$${L_1}:\overrightarrow r = \lambda \left( {\widehat i + 2\widehat j + 3\widehat k} \right),\,\lambda \in R$$

$${L_2}:\overrightarrow r = \left( {\widehat i + 3\widehat j + \widehat k} \right) + \mu \left( {\widehat i + \widehat j + 5\widehat k} \right);\,\mu \in R$$,

intersect at the point S. If a plane ax + by $$-$$ z + d = 0 passes through S and is parallel to both the lines L1 and L2, then the value of a + b + d is equal to ____________.