1
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
$$54+90 \sqrt{2}$$
2
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
$$\left(\frac{12}{7}, \infty\right)$$
3
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

Let the vectors $$\vec{a}=(1+t) \hat{i}+(1-t) \hat{j}+\hat{k}, \vec{b}=(1-t) \hat{i}+(1+t) \hat{j}+2 \hat{k}$$ and $$\vec{c}=t \hat{i}-t \hat{j}+\hat{k}, t \in \mathbf{R}$$ be such that for $$\alpha, \beta, \gamma \in \mathbf{R}, \alpha \vec{a}+\beta \vec{b}+\gamma \vec{c}=\overrightarrow{0} \Rightarrow \alpha=\beta=\gamma=0$$. Then, the set of all values of $$t$$ is :

A
a non-empty finite set
B
equal to $$\mathbf{N}$$
C
equal to $$\mathbf{R}-\{0\}$$
D
equal to $$\mathbf{R}$$
4
JEE Main 2022 (Online) 28th July Morning Shift
+4
-1

Let a vector $$\vec{a}$$ has magnitude 9. Let a vector $$\vec{b}$$ be such that for every $$(x, y) \in \mathbf{R} \times \mathbf{R}-\{(0,0)\}$$, the vector $$(x \vec{a}+y \vec{b})$$ is perpendicular to the vector $$(6 y \vec{a}-18 x \vec{b})$$. Then the value of $$|\vec{a} \times \vec{b}|$$ is equal to :

A
$$9 \sqrt{3}$$
B
$$27 \sqrt{3}$$
C
9
D
81
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