1
JEE Main 2024 (Online) 8th April Morning Shift
+4
-1

The set of all $$\alpha$$, for which the vectors $$\vec{a}=\alpha t \hat{i}+6 \hat{j}-3 \hat{k}$$ and $$\vec{b}=t \hat{i}-2 \hat{j}-2 \alpha t \hat{k}$$ are inclined at an obtuse angle for all $$t \in \mathbb{R}$$, is

A
$$[0,1)$$
B
$$\left(-\frac{4}{3}, 0\right]$$
C
$$(-2,0]$$
D
$$\left(-\frac{4}{3}, 1\right)$$
2
JEE Main 2024 (Online) 6th April Evening Shift
+4
-1

Let $$\vec{a}=2 \hat{i}+\hat{j}-\hat{k}, \vec{b}=((\vec{a} \times(\hat{i}+\hat{j})) \times \hat{i}) \times \hat{i}$$. Then the square of the projection of $$\vec{a}$$ on $$\vec{b}$$ is:

A
$$\frac{1}{3}$$
B
$$\frac{1}{5}$$
C
2
D
$$\frac{2}{3}$$
3
JEE Main 2024 (Online) 6th April Evening Shift
+4
-1

Let $$\overrightarrow{\mathrm{a}}=6 \hat{i}+\hat{j}-\hat{k}$$ and $$\overrightarrow{\mathrm{b}}=\hat{i}+\hat{j}$$. If $$\overrightarrow{\mathrm{c}}$$ is a is vector such that $$|\overrightarrow{\mathrm{c}}| \geq 6, \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{c}}=6|\overrightarrow{\mathrm{c}}|,|\overrightarrow{\mathrm{c}}-\overrightarrow{\mathrm{a}}|=2 \sqrt{2}$$ and the angle between $$\vec{a} \times \vec{b}$$ and $$\vec{c}$$ is $$60^{\circ}$$, then $$|(\vec{a} \times \vec{b}) \times \vec{c}|$$ is equal to:

A
$$\frac{3}{2} \sqrt{6}$$
B
$$\frac{9}{2}(6-\sqrt{6})$$
C
$$\frac{9}{2}(6+\sqrt{6})$$
D
$$\frac{3}{2} \sqrt{3}$$
4
JEE Main 2024 (Online) 5th April Evening Shift
+4
-1

Let $$\vec{a}=2 \hat{i}+5 \hat{j}-\hat{k}, \vec{b}=2 \hat{i}-2 \hat{j}+2 \hat{k}$$ and $$\vec{c}$$ be three vectors such that $$(\vec{c}+\hat{i}) \times(\vec{a}+\vec{b}+\hat{i})=\vec{a} \times(\vec{c}+\hat{i})$$. If $$\vec{a} \cdot \vec{c}=-29$$, then $$\vec{c} \cdot(-2 \hat{i}+\hat{j}+\hat{k})$$ is equal to:

A
15
B
10
C
5
D
12
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