1
JEE Main 2023 (Online) 8th April Evening Shift
+4
-1 The area of the quadrilateral $$\mathrm{ABCD}$$ with vertices $$\mathrm{A}(2,1,1), \mathrm{B}(1,2,5), \mathrm{C}(-2,-3,5)$$ and $$\mathrm{D}(1,-6,-7)$$ is equal to :

A
48
B
$$8 \sqrt{38}$$
C
54
D
$$9 \sqrt{38}$$
2
JEE Main 2023 (Online) 8th April Morning Shift
+4
-1 If the points with position vectors $$\alpha \hat{i}+10 \hat{j}+13 \hat{k}, 6 \hat{i}+11 \hat{j}+11 \hat{k}, \frac{9}{2} \hat{i}+\beta \hat{j}-8 \hat{k}$$ are collinear, then $$(19 \alpha-6 \beta)^{2}$$ is equal to :

A
16
B
49
C
36
D
25
3
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1 Let the vectors $$\vec{a}, \vec{b}, \vec{c}$$ represent three coterminous edges of a parallelopiped of volume V. Then the volume of the parallelopiped, whose coterminous edges are represented by $$\vec{a}, \vec{b}+\vec{c}$$ and $$\vec{a}+2 \vec{b}+3 \vec{c}$$ is equal to :

A
3 V
B
2 V
C
6 V
D
V
4
JEE Main 2023 (Online) 6th April Evening Shift
+4
-1 The sum of all values of $$\alpha$$, for which the points whose position vectors are $$\hat{i}-2 \hat{j}+3 \hat{k}, 2 \hat{i}-3 \hat{j}+4 \hat{k},(\alpha+1) \hat{i}+2 \hat{k}$$ and $$9 \hat{i}+(\alpha-8) \hat{j}+6 \hat{k}$$ are coplanar, is equal to :

A
6
B
4
C
$$-$$2
D
2
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