1
JEE Main 2024 (Online) 30th January Morning Shift
+4
-1

Let $$A(2,3,5)$$ and $$C(-3,4,-2)$$ be opposite vertices of a parallelogram $$A B C D$$. If the diagonal $$\overrightarrow{\mathrm{BD}}=\hat{i}+2 \hat{j}+3 \hat{k}$$, then the area of the parallelogram is equal to :

A
$$\frac{1}{2} \sqrt{410}$$
B
$$\frac{1}{2} \sqrt{306}$$
C
$$\frac{1}{2} \sqrt{586}$$
D
$$\frac{1}{2} \sqrt{474}$$
2
JEE Main 2024 (Online) 29th January Evening Shift
+4
-1

Let $$\mathrm{P}(3,2,3), \mathrm{Q}(4,6,2)$$ and $$\mathrm{R}(7,3,2)$$ be the vertices of $$\triangle \mathrm{PQR}$$. Then, the angle $$\angle \mathrm{QPR}$$ is

A
$$\cos ^{-1}\left(\frac{7}{18}\right)$$
B
$$\frac{\pi}{6}$$
C
$$\cos ^{-1}\left(\frac{1}{18}\right)$$
D
$$\frac{\pi}{3}$$
3
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
$$\frac{3}{2} \sqrt{31}$$
4
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
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