1
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1
Out of Syllabus

Let the position vectors of the points A, B, C and D be $$5 \hat{i}+5 \hat{j}+2 \lambda \hat{k}, \hat{i}+2 \hat{j}+3 \hat{k},-2 \hat{i}+\lambda \hat{j}+4 \hat{k}$$ and $$-\hat{i}+5 \hat{j}+6 \hat{k}$$. Let the set $$S=\{\lambda \in \mathbb{R}$$ : the points A, B, C and D are coplanar $$\}$$.

Then $$\sum_\limits{\lambda \in S}(\lambda+2)^{2}$$ is equal to :

A
$$\frac{37}{2}$$
B
25
C
13
D
41
2
JEE Main 2023 (Online) 6th April Morning Shift
+4
-1

Let $$\vec{a}=2 \hat{i}+3 \hat{j}+4 \hat{k}, \vec{b}=\hat{i}-2 \hat{j}-2 \hat{k}$$ and $$\vec{c}=-\hat{i}+4 \hat{j}+3 \hat{k}$$. If $$\vec{d}$$ is a vector perpendicular to both $$\vec{b}$$ and $$\vec{c}$$, and $$\vec{a} \cdot \vec{d}=18$$, then $$|\vec{a} \times \vec{d}|^{2}$$ is equal to :

A
680
B
720
C
760
D
640
3
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$\vec{a}=5 \hat{i}-\hat{j}-3 \hat{k}$$ and $$\vec{b}=\hat{i}+3 \hat{j}+5 \hat{k}$$ be two vectors. Then which one of the following statements is TRUE ?

A
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
B
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is opposite to the direction of $$\vec{b}$$.
C
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
D
Projection of $$\vec{a}$$ on $$\vec{b}$$ is $$\frac{-13}{\sqrt{35}}$$ and the direction of the projection vector is same as of $$\vec{b}$$.
4
JEE Main 2023 (Online) 1st February Evening Shift
+4
-1

Let $$\vec{a}=2 \hat{i}-7 \hat{j}+5 \hat{k}, \vec{b}=\hat{i}+\hat{k}$$ and $$\vec{c}=\hat{i}+2 \hat{j}-3 \hat{k}$$ be three given vectors. If $$\overrightarrow{\mathrm{r}}$$ is a vector such that $$\vec{r} \times \vec{a}=\vec{c} \times \vec{a}$$ and $$\vec{r} \cdot \vec{b}=0$$, then $$|\vec{r}|$$ is equal to :

A
$$\frac{11}{7}$$
B
$$\frac{11}{5} \sqrt{2}$$
C
$$\frac{\sqrt{914}}{7}$$
D
$$\frac{11}{7} \sqrt{2}$$
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