1
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

Let $$O$$ be the origin and the position vectors of $$A$$ and $$B$$ be $$2 \hat{i}+2 \hat{j}+\hat{k}$$ and $$2 \hat{i}+4 \hat{j}+4 \hat{k}$$ respectively. If the internal bisector of $$\angle \mathrm{AOB}$$ meets the line $$\mathrm{AB}$$ at $$\mathrm{C}$$, then the length of $$O C$$ is

A
$$\frac{3}{2} \sqrt{34}$$
B
$$\frac{2}{3} \sqrt{31}$$
C
$$\frac{2}{3} \sqrt{34}$$
D
$$\frac{3}{2} \sqrt{31}$$
2
JEE Main 2024 (Online) 29th January Morning Shift
+4
-1

Let $$P Q R$$ be a triangle with $$R(-1,4,2)$$. Suppose $$M(2,1,2)$$ is the mid point of $$\mathrm{PQ}$$. The distance of the centroid of $$\triangle \mathrm{PQR}$$ from the point of intersection of the lines $$\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}$$ and $$\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}$$ is

A
69
B
$$\sqrt{99}$$
C
$$\sqrt{69}$$
D
9
3
JEE Main 2024 (Online) 27th January Evening Shift
+4
-1

Let the image of the point $$(1,0,7)$$ in the line $$\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$$ be the point $$(\alpha, \beta, \gamma)$$. Then which one of the following points lies on the line passing through $$(\alpha, \beta, \gamma)$$ and making angles $$\frac{2 \pi}{3}$$ and $$\frac{3 \pi}{4}$$ with $$y$$-axis and $$z$$-axis respectively and an acute angle with $$x$$-axis ?

A
$$(1,-2,1+\sqrt{2})$$
B
$$(3,-4,3+2 \sqrt{2})$$
C
$$(3,4,3-2 \sqrt{2})$$
D
$$(1,2,1-\sqrt{2})$$
4
JEE Main 2024 (Online) 27th January Morning Shift
+4
-1
The distance, of the point $(7,-2,11)$ from the line

$\frac{x-6}{1}=\frac{y-4}{0}=\frac{z-8}{3}$ along the line $\frac{x-5}{2}=\frac{y-1}{-3}=\frac{z-5}{6}$, is :
A
12
B
18
C
21
D
14
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