Question 1: Palette 1/25

JEE Mains 2024 MCQ

The angle between vector $\vec{Q}$ and the resultant of $(2 \vec{Q}+2 \vec{P})$ and $(2 \vec{Q}-2 \vec{P})$ is :

A

$ \tan ^{-1}(\mathrm{P} / \mathrm{Q}) $

B

0$^\circ$

C

$ \tan ^{-1} \frac{(2 \vec{Q}-2 \vec{P})}{2 \vec{Q}+2 \vec{P}} $

D

$ \tan ^{-1}(2 Q / \mathrm{P}) $

Question 2: Palette 2/25

JEE Mains 2024 MCQ

If two vectors $\vec{A}$ and $\vec{B}$ having equal magnitude $R$ are inclined at angle $\theta$, then

A

$|\vec{A}+\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)$

B

$|\vec{A}-\vec{B}|=2 R \cos \left(\frac{\theta}{2}\right)$

C

$|\vec{A}-\vec{B}|=\sqrt{2} R \sin \left(\frac{\theta}{2}\right)$

D

$|\vec{A}+\vec{B}|=2 R \sin \left(\frac{\theta}{2}\right)$

Question 3: Palette 3/25

JEE Mains 2023 MCQ

A vector in $x-y$ plane makes an angle of $30^{\circ}$ with $y$-axis. The magnitude of $\mathrm{y}$-component of vector is $2 \sqrt{3}$. The magnitude of $x$-component of the vector will be :

A

$\sqrt{3}$

B

2

C

6

D

$\frac{1}{\sqrt{3}}$

Question 4: Palette 4/25

JEE Mains 2023 MCQ

When vector $\vec{A}=2 \hat{i}+3 \hat{j}+2 \hat{k}$ is subtracted from vector $\overrightarrow{\mathrm{B}}$, it gives a vector equal to $2 \hat{j}$. Then the magnitude of vector $\overrightarrow{\mathrm{B}}$ will be :

A

3

B

$\sqrt{33}$

C

$\sqrt6$

D

$\sqrt5$

Question 5: Palette 5/25

JEE Mains 2023 MCQ

Two forces having magnitude $A$ and $\frac{A}{2}$ are perpendicular to each other. The magnitude of their resultant is:

A

$\frac{5 A}{2}$

B

$\frac{\sqrt{5} A}{4}$

C

$\frac{\sqrt{5} A}{2}$

D

$\frac{\sqrt{5} A^{2}}{2}$

Question 6: Palette 6/25

JEE Mains 2023 MCQ

If two vectors $\overrightarrow P = \widehat i + 2m\widehat j + m\widehat k$ and $\overrightarrow Q = 4\widehat i - 2\widehat j + m\widehat k$ are perpendicular to each other. Then, the value of m will be :

A

$-1$

B

3

C

1

D

2

Question 7: Palette 7/25

JEE Mains 2022 MCQ

Two vectors $\overrightarrow A $ and $\overrightarrow B $ have equal magnitudes. If magnitude of $\overrightarrow A $ + $\overrightarrow B $ is equal to two times the magnitude of $\overrightarrow A $ $-$ $\overrightarrow B $, then the angle between $\overrightarrow A $ and $\overrightarrow B $ will be :

A

${\sin ^{ - 1}}\left( {{3 \over 5}} \right)$

B

${\sin ^{ - 1}}\left( {{1 \over 3}} \right)$

C

${\cos ^{ - 1}}\left( {{3 \over 5}} \right)$

D

${\cos ^{ - 1}}\left( {{1 \over 3}} \right)$

Question 8: Palette 8/25

JEE Mains 2022 MCQ

$\overrightarrow A $ is a vector quantity such that $|\overrightarrow A |$ = non-zero constant. Which of the following expression is true for $\overrightarrow A $ ?

A

$\overrightarrow A \,.\,\overrightarrow A = 0$

B

$\overrightarrow A \times \overrightarrow A < 0$

C

$\overrightarrow A \times \overrightarrow A = 0$

D

$\overrightarrow A \times \overrightarrow A > 0$

Question 9: Palette 9/25

JEE Mains 2022 MCQ

Which of the following relations is true for two unit vector $\widehat A$ and $\widehat B$ making an angle $\theta$ to each other?

A

$|\widehat A + \widehat B| = |\widehat A - \widehat B|\tan {\theta \over 2}$

B

$|\widehat A - \widehat B| = |\widehat A + \widehat B|\tan {\theta \over 2}$

C

$|\widehat A + \widehat B| = |\widehat A - \widehat B|cos{\theta \over 2}$

D

$|\widehat A - \widehat B| = |\widehat A + \widehat B|\cos {\theta \over 2}$

Question 10: Palette 10/25

JEE Mains 2021 MCQ

Statement I :

Two forces $\left( {\overrightarrow P + \overrightarrow Q } \right)$ and $\left( {\overrightarrow P - \overrightarrow Q } \right)$ where $\overrightarrow P \bot \overrightarrow Q $, when act at an angle $\theta$₁ to each other, the magnitude of their resultant is $\sqrt {3({P^2} + {Q^2})} $, when they act at an angle $\theta$₂, the magnitude of their resultant becomes $\sqrt {2({P^2} + {Q^2})} $. This is possible only when ${\theta _1} < {\theta _2}$.

Statement II :

In the situation given above.

$\theta$₁ = 60$^\circ$ and $\theta$₂ = 90$^\circ$

In the light of the above statements, choose the most appropriate answer from the options given below :-

A

Statement I is false but Statement II is true

B

Both Statement I and Statement II are true

C

Statement I is true but Statement II is false

D

Both Statement I and Statement II are false.

Question 11: Palette 11/25

JEE Mains 2021 MCQ

The resultant of these forces $\overrightarrow {OP} ,\overrightarrow {OQ} ,\overrightarrow {OR} ,\overrightarrow {OS} $ and $\overrightarrow {OT} $ is approximately .......... N.

[Take $\sqrt 3 = 1.7$, $\sqrt 2 = 1.4$ Given $\widehat i$ and $\widehat j$ unit vectors along x, y axis]

A

$9.25\widehat i + 5\widehat j$

B

$3\widehat i + 15\widehat j$

C

$2.5\widehat i - 14.5\widehat j$

D

$ - 1.5\widehat i - 15.5\widehat j$

Question 12: Palette 12/25

JEE Mains 2021 MCQ

The angle between vector $\left( {\overrightarrow A } \right)$ and $\left( {\overrightarrow A - \overrightarrow B } \right)$ is :

A

${\tan ^{ - 1}}\left( {{{ - {B \over 2}} \over {A - B{{\sqrt 3 } \over 2}}}} \right)$

B

${\tan ^{ - 1}}\left( {{A \over {0.7B}}} \right)$

C

${\tan ^{ - 1}}\left( {{{\sqrt 3 B} \over {2A - B}}} \right)$

D

${\tan ^{ - 1}}\left( {{{B\cos \theta } \over {A - B\sin \theta }}} \right)$

Question 13: Palette 13/25

JEE Mains 2021 MCQ

The magnitude of vectors $\overrightarrow {OA} $, $\overrightarrow {OB} $ and $\overrightarrow {OC} $ in the given figure are equal. The direction of $\overrightarrow {OA} $ + $\overrightarrow {OB} $ $-$ $\overrightarrow {OC} $ with x-axis will be :

A

${\tan ^{ - 1}}{{(1 - \sqrt 3 - \sqrt 2 )} \over {(1 + \sqrt 3 + \sqrt 2 )}}$

B

${\tan ^{ - 1}}{{(\sqrt 3 - 1 + \sqrt 2 )} \over {(1 + \sqrt 3 - \sqrt 2 )}}$

C

${\tan ^{ - 1}}{{(\sqrt 3 - 1 + \sqrt 2 )} \over {(1 - \sqrt 3 + \sqrt 2 )}}$

D

${\tan ^{ - 1}}{{(1 + \sqrt 3 - \sqrt 2 )} \over {(1 - \sqrt 3 - \sqrt 2 )}}$

Question 14: Palette 14/25

JEE Mains 2021 MCQ

Assertion A : If A, B, C, D are four points on a semi-circular are with centre at 'O' such that $\left| {\overrightarrow {AB} } \right| = \left| {\overrightarrow {BC} } \right| = \left| {\overrightarrow {CD} } \right|$, then $\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} = 4\overrightarrow {AO} + \overrightarrow {OB} + \overrightarrow {OC} $

Reason R : Polygon law of vector addition yields $\overrightarrow {AB} + \overrightarrow {BC} + \overrightarrow {CD} + \overrightarrow {AD} = 2\overrightarrow {AO} $


In the light of the above statements, choose the most appropriate answer from the options given below :

A

A is correct but R is not correct.

B

A is not correct but R is correct.

C

Both A and R are correct and R is the correct explanation of A.

D

Both A and R are correct but R is not the correct explanation of A.

Question 15: Palette 15/25

JEE Mains 2021 MCQ

Two vectors $\overrightarrow X $ and $\overrightarrow Y $ have equal magnitude. The magnitude of ($\overrightarrow X $ $-$ $\overrightarrow Y $) is n times the magnitude of ($\overrightarrow X $ + $\overrightarrow Y $). The angle between $\overrightarrow X $ and $\overrightarrow Y $ is :

A

${\cos ^{ - 1}}\left( {{{ - {n^2} - 1} \over {{n^2} - 1}}} \right)$

B

${\cos ^{ - 1}}\left( {{{{n^2} - 1} \over { - {n^2} - 1}}} \right)$

C

${\cos ^{ - 1}}\left( {{{{n^2} + 1} \over { - {n^2} - 1}}} \right)$

D

${\cos ^{ - 1}}\left( {{{{n^2} + 1} \over {{n^2} - 1}}} \right)$

Question 16: Palette 16/25

JEE Mains 2021 MCQ

Match List - I with List - II


Choose the correct answer from the options given below :

A

(a) $\to$ (iv), (b) $\to$ (i), (c) $\to$ (iii), (d) $\to$ (ii)

B

(a) $\to$ (iv), (b) $\to$ (iii), (c) $\to$ (i), (d) $\to$ (ii)

C

(a) $\to$ (iii), (b) $\to$ (ii), (c) $\to$ (iv), (d) $\to$ (i)

D

(a) $\to$ (i), (b) $\to$ (iv), (c) $\to$ (ii), (d) $\to$ (iii)

Question 17: Palette 17/25

JEE Mains 2021 MCQ

What will be the projection of vector $\overrightarrow A = \widehat i + \widehat j + \widehat k$ on vector $\overrightarrow B = \widehat i + \widehat j$ ?

A

$\sqrt 2 (\widehat i + \widehat j + \widehat k)$

B

$(\widehat i + \widehat j)$

C

$\sqrt 2 (\widehat i + \widehat j)$

D

$2(\widehat i + \widehat j + \widehat k)$

Question 18: Palette 18/25

JEE Mains 2021 MCQ

Two vectors ${\overrightarrow P }$ and ${\overrightarrow Q }$ have equal magnitudes. If the magnitude of ${\overrightarrow P + \overrightarrow Q }$ is n times the magnitude of ${\overrightarrow P - \overrightarrow Q }$, then angle between ${\overrightarrow P }$ and ${\overrightarrow Q }$ is :

A

${\sin ^{ - 1}}\left( {{{n - 1} \over {n + 1}}} \right)$

B

${\cos ^{ - 1}}\left( {{{n - 1} \over {n + 1}}} \right)$

C

${\sin ^{ - 1}}\left( {{{{n^2} - 1} \over {{n^2} + 1}}} \right)$

D

${\cos ^{ - 1}}\left( {{{{n^2} - 1} \over {{n^2} + 1}}} \right)$

Question 19: Palette 19/25

JEE Mains 2021 MCQ

If $\overrightarrow A $ and $\overrightarrow B $ are two vectors satisfying the relation $\overrightarrow A $ . $\overrightarrow B $ = $\left| {\overrightarrow A \times \overrightarrow B } \right|$. Then the value of $\left| {\overrightarrow A - \overrightarrow B } \right|$ will be :

A

$\sqrt {{A^2} + {B^2} + \sqrt 2 AB} $

B

$\sqrt {{A^2} + {B^2}} $

C

$\sqrt {{A^2} + {B^2} - \sqrt 2 AB} $

D

$\sqrt {{A^2} + {B^2} + 2AB} $

Question 20: Palette 20/25

JEE Mains 2021 MCQ

In an octagon ABCDEFGH of equal side, what is the sum of

$\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} + \overrightarrow {AE} + \overrightarrow {AF} + \overrightarrow {AG} + \overrightarrow {AH} $,

if, $\overrightarrow {AO} = 2\widehat i + 3\widehat j - 4\widehat k$

A

$ - 16\widehat i - 24\widehat j + 32\widehat k$

B

$16\widehat i + 24\widehat j - 32\widehat k$

C

$16\widehat i + 24\widehat j + 32\widehat k$

D

$16\widehat i - 24\widehat j + 32\widehat k$

Question 21: Palette 21/25

JEE Mains 2019 MCQ

Let $\left| {\mathop {{A_1}}\limits^ \to } \right| = 3$, $\left| {\mathop {{A_2}}\limits^ \to } \right| = 5$ and $\left| {\mathop {{A_1}}\limits^ \to + \mathop {{A_2}}\limits^ \to } \right| = 5$. The value of $\left( {2\mathop {{A_1}}\limits^ \to + 3\mathop {{A_2}}\limits^ \to } \right)\left( {3\mathop {{A_1}}\limits^ \to - \mathop {2{A_2}}\limits^ \to } \right)$ is :-

A

–118.5

B

–112.5

C

–99.5

D

–106.5

Question 22: Palette 22/25

JEE Mains 2019 MCQ

Two vectors $\overrightarrow A $ and $\overrightarrow B $ have equal magnitudes. The magnitude of $\left( {\overrightarrow A + \overrightarrow B } \right)$ is 'n' times the magnitude of $\left( {\overrightarrow A - \overrightarrow B } \right)$ . The angle between ${\overrightarrow A }$ and ${\overrightarrow B }$ is -

A

${\sin ^{ - 1}}\left[ {{{n - 1} \over {n + 1}}} \right]$

B

${\sin ^{ - 1}}\left[ {{{{n^2} - 1} \over {{n^2} + 1}}} \right]$

C

${\cos ^{ - 1}}\left[ {{{{n^2} - 1} \over {{n^2} + 1}}} \right]$

D

${\cos ^{ - 1}}\left[ {{{n - 1} \over {n + 1}}} \right]$

Question 23: Palette 23/25

JEE Mains 2019 MCQ

In the cube of side ‘a’ shown in the figure, the vector from the central point of the face ABOD to the central point of the face BEFO will be -

A

${1 \over 2}a\left( {\widehat k - \widehat i} \right)$

B

${1 \over 2}a\left( {\widehat j - \widehat i} \right)$

C

${1 \over 2}a\left( {\widehat j - \widehat k} \right)$

D

${1 \over 2}a\left( {\widehat i - \widehat k} \right)$

Question 24: Palette 24/25

JEE Mains 2018 MCQ

Let $\overrightarrow A $ = $\left( {\widehat i + \widehat j} \right)$ and, $\overrightarrow B = \left( {2\widehat i - \widehat j} \right).$ The magnitude of a coplanar vector $\overrightarrow C $ such that $\overrightarrow A .\overrightarrow C = \overrightarrow B .\overrightarrow C = \overrightarrow A .\overrightarrow B ,$ is given by :

A

$\sqrt {{{10} \over 9}} $

B

$\sqrt {{{5} \over 9}} $

C

$\sqrt {{{20} \over 9}} $

D

$\sqrt {{{9} \over 12}} $

Question 25: Palette 25/25

JEE Mains 2004 MCQ

If $\overrightarrow A \times \overrightarrow B = \overrightarrow B \times \overrightarrow A $, then the angle beetween A and B is

A

${\pi \over 2}$

B

${\pi \over 3}$

C

$\pi $

D

${\pi \over 4}$