1
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1 If the equation of the plane passing through the line of intersection of the planes $$2 x-y+z=3,4 x-3 y+5 z+9=0$$ and parallel to the line $$\frac{x+1}{-2}=\frac{y+3}{4}=\frac{z-2}{5}$$ is $$a x+b y+c z+6=0$$, then $$a+b+c$$ is equal to :

A
13
B
15
C
14
D
12
2
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1 One vertex of a rectangular parallelopiped is at the origin $$\mathrm{O}$$ and the lengths of its edges along $$x, y$$ and $$z$$ axes are $$3,4$$ and $$5$$ units respectively. Let $$\mathrm{P}$$ be the vertex $$(3,4,5)$$. Then the shortest distance between the diagonal OP and an edge parallel to $$\mathrm{z}$$ axis, not passing through $$\mathrm{O}$$ or $$\mathrm{P}$$ is :

A
$$\frac{12}{\sqrt{5}}$$
B
$$12 \sqrt{5}$$
C
$$\frac{12}{5}$$
D
$$\frac{12}{5 \sqrt{5}}$$
3
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1 Let the plane P pass through the intersection of the planes $$2x+3y-z=2$$ and $$x+2y+3z=6$$, and be perpendicular to the plane $$2x+y-z+1=0$$. If d is the distance of P from the point ($$-$$7, 1, 1), then $$\mathrm{d^{2}}$$ is equal to :

A
$$\frac{250}{83}$$
B
$$\frac{250}{82}$$
C
$$\frac{15}{53}$$
D
$$\frac{25}{83}$$
4
JEE Main 2023 (Online) 1st February Morning Shift
+4
-1 The shortest distance between the lines

$${{x - 5} \over 1} = {{y - 2} \over 2} = {{z - 4} \over { - 3}}$$ and

$${{x + 3} \over 1} = {{y + 5} \over 4} = {{z - 1} \over { - 5}}$$ is :

A
$$7\sqrt 3$$
B
$$5\sqrt 3$$
C
$$4\sqrt 3$$
D
$$6\sqrt 3$$
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