1
JEE Main 2022 (Online) 29th July Evening Shift
+4
-1 Let $$\vec{a}, \vec{b}, \vec{c}$$ be three coplanar concurrent vectors such that angles between any two of them is same. If the product of their magnitudes is 14 and $$(\vec{a} \times \vec{b}) \cdot(\vec{b} \times \vec{c})+(\vec{b} \times \vec{c}) \cdot(\vec{c} \times \vec{a})+(\vec{c} \times \vec{a}) \cdot(\vec{a} \times \vec{b})=168$$, then $$|\vec{a}|+|\vec{b}|+|\vec{c}|$$ is equal to :

A
10
B
14
C
16
D
18
2
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1 Let $$\overrightarrow{\mathrm{a}}=3 \hat{i}+\hat{j}$$ and $$\overrightarrow{\mathrm{b}}=\hat{i}+2 \hat{j}+\hat{k}$$. Let $$\overrightarrow{\mathrm{c}}$$ be a vector satisfying $$\overrightarrow{\mathrm{a}} \times(\overrightarrow{\mathrm{b}} \times \overrightarrow{\mathrm{c}})=\overrightarrow{\mathrm{b}}+\lambda \overrightarrow{\mathrm{c}}$$. If $$\overrightarrow{\mathrm{b}}$$ and $$\overrightarrow{\mathrm{c}}$$ are non-parallel, then the value of $$\lambda$$ is:

A
$$-$$5
B
5
C
1
D
$$-$$1
3
JEE Main 2022 (Online) 29th July Morning Shift
+4
-1 Let $$\hat{a}$$ and $$\hat{b}$$ be two unit vectors such that the angle between them is $$\frac{\pi}{4}$$. If $$\theta$$ is the angle between the vectors $$(\hat{a}+\hat{b})$$ and $$(\hat{a}+2 \hat{b}+2(\hat{a} \times \hat{b}))$$, then the value of $$164 \,\cos ^{2} \theta$$ is equal to :

A
$$90+27 \sqrt{2}$$
B
$$45+18 \sqrt{2}$$
C
$$90+3 \sqrt{2}$$
D
$$54+90 \sqrt{2}$$
4
JEE Main 2022 (Online) 28th July Evening Shift
+4
-1 Let S be the set of all a $$\in R$$ for which the angle between the vectors $$\vec{u}=a\left(\log _{e} b\right) \hat{i}-6 \hat{j}+3 \hat{k}$$ and $$\vec{v}=\left(\log _{e} b\right) \hat{i}+2 \hat{j}+2 a\left(\log _{e} b\right) \hat{k}$$, $$(b>1)$$ is acute. Then S is equal to :

A
$$\left(-\infty,-\frac{4}{3}\right)$$
B
$$\Phi$$
C
$$\left(-\frac{4}{3}, 0\right)$$
D
$$\left(\frac{12}{7}, \infty\right)$$
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