1
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1 Let $S$ be the set of all $(\lambda, \mu)$ for which the vectors $\lambda \hat{i}-\hat{j}+\hat{k}, \hat{i}+2 \hat{j}+\mu \hat{k}$ and $3 \hat{i}-4 \hat{j}+5 \hat{k}$, where $\lambda-\mu=5$, are coplanar, then $\sum\limits_{(\lambda, \mu) \in S} 80\left(\lambda^2+\mu^2\right)$ is equal to :
A
2370
B
2130
C
2210
D
2290
2
JEE Main 2023 (Online) 15th April Morning Shift
+4
-1 Let $\mathrm{ABCD}$ be a quadrilateral. If $\mathrm{E}$ and $\mathrm{F}$ are the mid points of the diagonals $\mathrm{AC}$ and $\mathrm{BD}$ respectively and $(\overrightarrow{A B}-\overrightarrow{B C})+(\overrightarrow{A D}-\overrightarrow{D C})=k \overrightarrow{F E}$, then $k$ is equal to :
A
-2
B
4
C
-4
D
2
3
JEE Main 2023 (Online) 13th April Evening Shift
+4
-1 Let $$|\vec{a}|=2,|\vec{b}|=3$$ and the angle between the vectors $$\vec{a}$$ and $$\vec{b}$$ be $$\frac{\pi}{4}$$. Then $$|(\vec{a}+2 \vec{b}) \times(2 \vec{a}-3 \vec{b})|^{2}$$ is equal to :

A
441
B
482
C
841
D
882
4
JEE Main 2023 (Online) 13th April Evening Shift
+4
-1 Let for a triangle $$\mathrm{ABC}$$,

$$\overrightarrow{\mathrm{AB}}=-2 \hat{i}+\hat{j}+3 \hat{k}$$

$$\overrightarrow{\mathrm{CB}}=\alpha \hat{i}+\beta \hat{j}+\gamma \hat{k}$$

$$\overrightarrow{\mathrm{CA}}=4 \hat{i}+3 \hat{j}+\delta \hat{k}$$

If $$\delta > 0$$ and the area of the triangle $$\mathrm{ABC}$$ is $$5 \sqrt{6}$$, then $$\overrightarrow{C B} \cdot \overrightarrow{C A}$$ is equal to

A
60
B
54
C
120
D
108
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