1
JEE Main 2025 (Online) 8th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $ \vec{a} = \hat{i} + 2\hat{j} + \hat{k} $ and $ \vec{b} = 2\hat{i} + \hat{j} - \hat{k} $. Let $ \hat{c} $ be a unit vector in the plane of the vectors $ \vec{a} $ and $ \vec{b} $ and be perpendicular to $ \vec{a} $. Then such a vector $ \hat{c} $ is:

A

$ \frac{1}{\sqrt{2}}(-\hat{i} + \hat{k}) $

B

$ \frac{1}{\sqrt{5}}(\hat{j} - 2\hat{k}) $

C

$ \frac{1}{\sqrt{3}}(\hat{i} - \hat{j} + \hat{k}) $

D

$ \frac{1}{\sqrt{3}}(-\hat{i} + \hat{j} - \hat{k}) $

2
JEE Main 2025 (Online) 7th April Evening Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let $ \vec{a} $ and $ \vec{b} $ be the vectors of the same magnitude such that

$ \frac{|\vec{a} + \vec{b}| + |\vec{a} - \vec{b}|}{|\vec{a} + \vec{b}| - |\vec{a} - \vec{b}|} = \sqrt{2} + 1. $ Then $ \frac{|\vec{a} + \vec{b}|^2}{|\vec{a}|^2} $ is :

A

2 + $\sqrt{2}$

B

2 + 4$\sqrt{2}$

C

4 + 2$\sqrt{2}$

D

1 + $\sqrt{2}$

3
JEE Main 2025 (Online) 7th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Let the angle $\theta, 0<\theta<\frac{\pi}{2}$ between two unit vectors $\hat{a}$ and $\hat{b}$ be $\sin ^{-1}\left(\frac{\sqrt{65}}{9}\right)$. If the vector $\vec{c}=3 \hat{a}+6 \hat{b}+9(\hat{a} \times \hat{b})$, then the value of $9(\vec{c} \cdot \hat{a})-3(\vec{c} \cdot \hat{b})$ is

A
31
B
29
C
24
D
27
4
JEE Main 2025 (Online) 4th April Morning Shift
MCQ (Single Correct Answer)
+4
-1
Change Language

Consider two vectors $\vec{u}=3 \hat{i}-\hat{j}$ and $\vec{v}=2 \hat{i}+\hat{j}-\lambda \hat{k}, \lambda>0$. The angle between them is given by $\cos ^{-1}\left(\frac{\sqrt{5}}{2 \sqrt{7}}\right)$. Let $\vec{v}=\vec{v}_1+\overrightarrow{v_2}$, where $\vec{v}_1$ is parallel to $\vec{u}$ and $\overrightarrow{v_2}$ is perpendicular to $\vec{u}$. Then the value $\left|\overrightarrow{v_1}\right|^2+\left|\overrightarrow{v_2}\right|^2$ is equal to

A
$\frac{23}{2}$
B
$\frac{25}{2}$
C
10
D
14
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