1
JEE Main 2023 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
A vector $\vec{v}$ in the first octant is inclined to the $x$-axis at $60^{\circ}$, to the $y$-axis at 45 and to the $z$-axis at an acute angle. If a plane passing through the points $(\sqrt{2},-1,1)$ and $(a, b, c)$, is normal to $\vec{v}$, then :
A
$a+b+\sqrt{2} c=1$
B
$\sqrt{2} a+b+c=1$
C
$\sqrt{2} a-b+c=1$
D
$a+\sqrt{2} b+c=1$
2
JEE Main 2023 (Online) 30th January Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language
If a plane passes through the points $(-1, k, 0),(2, k,-1),(1,1,2)$ and is parallel to the line $\frac{x-1}{1}=\frac{2 y+1}{2}=\frac{z+1}{-1}$, then the value of $\frac{k^2+1}{(k-1)(k-2)}$ is :
A
$\frac{17}{5}$
B
$\frac{6}{13}$
C
$\frac{13}{6}$
D
$\frac{5}{17}$
3
JEE Main 2023 (Online) 30th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

The line $$l_1$$ passes through the point (2, 6, 2) and is perpendicular to the plane $$2x+y-2z=10$$. Then the shortest distance between the line $$l_1$$ and the line $$\frac{x+1}{2}=\frac{y+4}{-3}=\frac{z}{2}$$ is :

A
9
B
7
C
$$\frac{19}{3}$$
D
$$\frac{13}{3}$$
4
JEE Main 2023 (Online) 30th January Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let a unit vector $$\widehat{O P}$$ make angles $$\alpha, \beta, \gamma$$ with the positive directions of the co-ordinate axes $$\mathrm{OX}$$, $$\mathrm{OY}, \mathrm{OZ}$$ respectively, where $$\beta \in\left(0, \frac{\pi}{2}\right)$$. If $$\widehat{\mathrm{OP}}$$ is perpendicular to the plane through points $$(1,2,3),(2,3,4)$$ and $$(1,5,7)$$, then which one of the following is true ?

A
$$\alpha \in\left(\frac{\pi}{2}, \pi\right)$$ and $$\gamma \in\left(\frac{\pi}{2}, \pi\right)$$
B
$$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and $$\gamma \in\left(\frac{\pi}{2}, \pi\right)$$
C
$$\alpha \in\left(\frac{\pi}{2}, \pi\right)$$ and $$\gamma \in\left(0, \frac{\pi}{2}\right)$$
D
$$\alpha \in\left(0, \frac{\pi}{2}\right)$$ and $$\gamma \in\left(0, \frac{\pi}{2}\right)$$
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