Question 1: Palette 1/53

DPT MCQ

Two forces $\vec{A}$ and $\vec{B}$ of magnitude 10 N and 20 N are acting on a block having 60° angle between them. Find the magnitude of resultant of them.

A

$10\sqrt{3}$ N

B

$10\sqrt{5}$ N

C

$10\sqrt{7}$ N

D

30 N

Question 2: Palette 2/53

DPT MCQ

Max and min magnitude of resultant of two forces are 7 N and 1 N respectively. Find the resultant of 2 forces when it act orthogonally.

A

3 N

B

4 N

C

5 N

D

7 N

Question 3: Palette 3/53

DPT MCQ

Two vectors A & B have the same magnitude x. Find the magnitude of the resultant of A and B if the angle between them is 60°.

A

x

B

$x\sqrt{3}$

C

$x\sqrt{2}$

D

2x

Question 4: Palette 4/53

DPT MCQ

If two vectors A and B of magnitude 10 N and 6 N are at an angle 60°. If B becomes twice its initial value and is added to A. Find the magnitude of the resultant of A and B after the change in B.

A

$\sqrt{244}$ N

B

$\sqrt{364}$ N

C

20 N

D

26 N

Question 5: Palette 5/53

DPT MCQ

Magnitude of vector A is 8 N. Magnitude of vector B is 6 N. Which of the following can be the magnitude of A + B?

A

10 N

B

22 N

C

48 N

D

1.99 N

Question 6: Palette 6/53

DPT MCQ

Magnitude of resultant of vectors A and B is 5 unit where magnitude of vector A is $5\sqrt{3}$ unit and magnitude of vector B is 5 unit. Find the angle between vectors A and B.

A

30°

B

60°

C

120°

D

150°

Question 7: Palette 7/53

DPT MCQ

Sum of the magnitude of vector A and vector B is 16 N. Magnitude of resultant of vector A and B is 8 N. When resultant is perpendicular to vector A. Find magnitude of A and B.

A

A = 6 N, B = 10 N

B

A = 10 N, B = 6 N

C

A = 8 N, B = 8 N

D

A = 4 N, B = 12 N

Question 8: Palette 8/53

DPT MCQ

Resultant of vector A and vector B is perpendicular to vector A and its magnitude is equal to half of the magnitude of vector B. Find the angle between vector A and vector B.

A

30°

B

60°

C

120°

D

150°

Question 9: Palette 9/53

DPT MCQ

Two vectors A and B have the same magnitude 'a' and their resultant has magnitude R. Now B is doubled and added to A and the new resultant becomes $a\sqrt{3}$. Find the angle between A and B.

A

60°

B

90°

C

120°

D

150°

Question 10: Palette 10/53

DPT MCQ

If magnitude of resultant of 2N and 3N is 4N find angle between 2N and 3N:

A

$cos^{-1}(1/4)$

B

$cos^{-1}(1/3)$

C

$cos^{-1}(1/2)$

D

$cos^{-1}(3/4)$

Question 11: Palette 11/53

DPT MCQ

Two vectors of magnitude 4N and 6N are acting at an angle $60^{\circ}$ then find:
(i) Magnitude of their resultant vectors
(ii) Angle between resultant vector and 4N (iii) Angle between resultant vector and 6N

A

(i) $2\sqrt{19}N$, (ii) $tan^{-1}(\frac{3\sqrt{3}}{7})$, (iii) $tan^{-1}(\frac{\sqrt{3}}{4})$

B

(i) $\sqrt{52}N$, (ii) $tan^{-1}(\frac{3\sqrt{3}}{7})$, (iii) $tan^{-1}(\frac{\sqrt{3}}{4})$

C

(i) $2\sqrt{19}N$, (ii) $tan^{-1}(\frac{2\sqrt{3}}{7})$, (iii) $tan^{-1}(\frac{\sqrt{3}}{5})$

D

(i) $10N$, (ii) $30^{\circ}$, (iii) $30^{\circ}$

Question 12: Palette 12/53

DPT MCQ

What can be resultant of two vectors $\vec{A}$ and $\vec{B}$ of magnitude 3 N and 5 N?

A

1 N, 10 N, 12 N

B

2 N, 5 N, 6 N

C

0 N, 1 N, 10 N

D

10 N, 12 N, 0 N

Question 13: Palette 13/53

DPT MCQ

If 100 coplanar vectors each having magnitude 10 units are equally inclined with each other then find the magnitude of their resultant?

A

10

B

100

C

1000

Question 14: Palette 14/53

DPT MCQ

Which of the following groups of vectors can give a zero resultant (Equilibrium)?

A

2N, 4N, 8N

B

10N, 12N, 23N

C

1N, 2N, 3N

D

1N, 8N, 2N

Question 15: Palette 15/53

DPT MCQ

It is a regular hexagon and O is the centre then find $\vec{OA} + \vec{OB} + \vec{OC} + \vec{OD} + \vec{OE} + \vec{OF} = ?$

A

$\vec{0}$

B

$6\vec{a}$

C

$3\vec{a}$

D

$\vec{a}$

Question 16: Palette 16/53

DPT MCQ

Find the angle between $\vec{A}$ and $\vec{B}$ if $|\vec{A} - \vec{B}| = |\vec{A} + \vec{B}|$

A

$90^{\circ}$

B

$60^{\circ}$

C

$0^{\circ}$

D

$180^{\circ}$

Question 17: Palette 17/53

DPT MCQ

The unit vector along $\hat{i}-2\hat{j}$ is:

A

$\frac{\hat{i}-2\hat{j}}{\sqrt{5}}$

B

$\hat{i}+\hat{j}$

C

$\frac{\hat{i}+\hat{j}}{\sqrt{2}}$

D

$\frac{\hat{i}-\hat{j}}{\sqrt{5}}$

Question 18: Palette 18/53

DPT MCQ

If a unit vector is represented by $0.3\hat{i}-0.4\hat{j}+c\hat{k}$, then the value of c is:

A

$\sqrt{0.75}$

B

$\sqrt{0.25}$

C

$\sqrt{0.01}$

D

$\sqrt{0.39}$

Question 19: Palette 19/53

DPT MCQ

A force $\vec{F}=(2\hat{i}+3\hat{j})\text{ N}$ acts on a body and displaces it by $\vec{S}=(3\hat{i}+4\hat{j})\text{ m}$. The work done $(W=\vec{F}\cdot\vec{S})$ by the force is:

A

10 J

B

12 J

C

18 J

D

25 J

Question 20: Palette 20/53

DPT MCQ

A force $\vec{F} = (2\hat{i}+2\hat{j})\text{ N}$ displaces an object through a distance $\vec{S} = (2\hat{i}-3\hat{j})\text{ m}$. The work done ($W=\vec{F}\cdot\vec{S}$) is:

A

-2 J

B

12 J

C

5 J

D

13 J

Question 21: Palette 21/53

DPT MCQ

uestion: A force of $14\text{ N}$ acts on a particle along the vector $(3\hat{i}+2\hat{j}-6\hat{k})$. If the particle displaces from $(0, 0, 0)$ to $(2, 4, -2)$, the work done $(W=\vec{F}\cdot\vec{S})$ by the force on the particle is:

A

10 J

B

52 J

C

-48 J

D

14 J

Question 22: Palette 22/53

DPT MCQ

If a vector $(2\hat{i}+3\hat{j}+8\hat{k})$ is perpendicular to the vector $(4\hat{i}-4\alpha\hat{j}+\alpha\hat{k})$, then the value of $\alpha$ is:

A

-1

B

1/2

C

-1/2

D

2

Question 23: Palette 23/53

DPT MCQ

If $\hat{n} = a\hat{i}+b\hat{j}$ is a unit vector perpendicular to the vector $(\hat{i}-\hat{j})$, then the value of a and b may be:

A

1, 0

B

-2, 0

C

3, 0

D

$\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}$

Question 24: Palette 24/53

DPT MCQ

The angle between the two vectors $\vec{A}=3\hat{i}+4\hat{j}+5\hat{k}$ and $\vec{B}=6\hat{i}+8\hat{j}+10\hat{k}$ will be:

A

$0^{\circ}$

B

$180^{\circ}$

C

$90^{\circ}$

D

$45^{\circ}$

Question 25: Palette 25/53

DPT MCQ

The angle between vectors $(\hat{i}+\hat{j})$ and $(\hat{i}+\hat{k})$ is:

A

$90^{\circ}$

B

$180^{\circ}$

C

$0^{\circ}$

D

$60^{\circ}$

Question 26: Palette 26/53

DPT MCQ

The angle between $2\hat{i}+\hat{j}+2\hat{k}$ and $\hat{i}-\hat{j}+\hat{k}$ is:

A

$30^{\circ}$

B

$60^{\circ}$

C

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

D

$\cos^{-1}\left(\frac{2}{3}\right)$

Question 27: Palette 27/53

DPT MCQ

What is the projection of $3\hat{i}+4\hat{k}$ on the z-axis?

A

3

B

4

C

5

D

zero

Question 28: Palette 28/53

DPT MCQ

What is the projection of vector $\vec{B}$ on vector $\vec{A}$?

A

$\vec{A} \cdot \vec{B}$

B

$\vec{A} \cdot \hat{B}$

C

$\vec{B} \cdot \hat{A}$

D

$\hat{A} \cdot \hat{B}$

Question 29: Palette 29/53

DPT MCQ

Given that $\vert{}\vec{A}\vert{} = \vert{}\vec{B}\vert{}$ and $\vec{A} \perp \vec{B}$. What is the angle between $(\vec{A}+\vec{B})$ and $(\vec{A}-\vec{B})$?

A

$30^{\circ}$

B

$60^{\circ}$

C

$90^{\circ}$

D

$180^{\circ}$

Question 30: Palette 30/53

DPT MCQ

A particle has momentum of magnitude $20\text{ kg}\cdot\text{m/s}$. If the momentum is in the direction of $\vec{A}$, find the momentum in vector form, where $\vec{A} = \hat{i} + \hat{j}$.

A

$10\hat{i} + 10\hat{j}\text{ kg}\cdot\text{m/s}$

B

$10\sqrt{2}\hat{i} + 10\sqrt{2}\hat{j}\text{ kg}\cdot\text{m/s}$

C

$20\hat{i} + 20\hat{j}\text{ kg}\cdot\text{m/s}$

D

$20\sqrt{2}\hat{i} + 20\sqrt{2}\hat{j}\text{ kg}\cdot\text{m/s}$

Question 31: Palette 31/53

DPT MCQ

Find the component of $\vec{A} = 3\hat{i} + 4\hat{j}$ along $\vec{B} = \hat{i} + \hat{j}$.

A

Scalar: $\frac{7}{\sqrt{2}}$, Vector: $\frac{7}{2}(\hat{i} + \hat{j})$

B

Scalar: $\frac{5}{\sqrt{2}}$, Vector: $\frac{5}{2}(\hat{i} + \hat{j})$

C

Scalar: $7$, Vector: $7(\hat{i} + \hat{j})$

D

Scalar: $\frac{7}{2}$, Vector: $\frac{7}{\sqrt{2}}(\hat{i} + \hat{j})$

Question 32: Palette 32/53

DPT MCQ

A bird is flying with speed $10\text{ m/s}$ in the direction of a vector $\vec{A} = 3\hat{i} + 4\hat{j}$. Find the velocity of the bird in vector form.

A

$3\hat{i} + 4\hat{j}\text{ m/s}$

B

$6\hat{i} + 8\hat{j}\text{ m/s}$

C

$10\hat{i} + 10\hat{j}\text{ m/s}$

D

$15\hat{i} + 20\hat{j}\text{ m/s}$

Question 33: Palette 33/53

DPT MCQ

If $\vec{A} = \hat{i} + 3\hat{j}$ and $\vec{B} = 3\hat{i} + 4\hat{j}$, find the component of $\vec{A}$ perpendicular to $\vec{B}$.

A

$\frac{-4\hat{i} + 3\hat{j}}{5}$

B

$\frac{-4\hat{i} + 3\hat{j}}{25}$

C

$\frac{-12\hat{i} + 16\hat{j}}{25}$

D

$\frac{-4\hat{i} - 3\hat{j}}{5}$

Question 34: Palette 34/53

DPT MCQ

For the vectors $\vec{A} = 3\hat{i} - 4\hat{j} + 5\hat{k}$ and $\vec{B} = -9\hat{i} + 12\hat{j} - 15\hat{k}$, determine the relationship between them.

A

The vectors are parallel

B

The vectors are anti-parallel

C

The vectors are perpendicular

D

The vectors are equal

Question 35: Palette 35/53

DPT MCQ

If $\vec{A} = 4\hat{i} - 2\hat{j}$ and $\vec{B} = 3\hat{i} + 4\hat{j}$, find the component of $\vec{A}$ perpendicular to $\vec{B}$.

A

$\frac{88\hat{i} - 66\hat{j}}{25}$

B

$\frac{88\hat{i} + 66\hat{j}}{25}$

C

$\frac{12\hat{i} + 16\hat{j}}{25}$

D

$\frac{4\hat{i} - 2\hat{j}}{5}$

Question 36: Palette 36/53

DPT MCQ

If $\vec{A} = 2\hat{i} + 3\hat{j} + 6\hat{k}$, find its direction cosines.

A

$\cos\alpha = \frac{2}{7}$, $\cos\beta = \frac{3}{7}$, $\cos\gamma = \frac{6}{7}$

B

$\cos\alpha = \frac{2}{\sqrt{11}}$, $\cos\beta = \frac{3}{\sqrt{11}}$, $\cos\gamma = \frac{6}{\sqrt{11}}$

C

$\cos\alpha = \frac{3}{7}$, $\cos\beta = \frac{2}{7}$, $\cos\gamma = \frac{6}{7}$

D

$\cos\alpha = \frac{1}{7}$, $\cos\beta = \frac{2}{7}$, $\cos\gamma = \frac{3}{7}$

Question 37: Palette 37/53

DPT MCQ

If $\vec{A} = 3\hat{i} + 4\hat{j} + 5\hat{k}$, find its direction cosines and components along the x, y, and z axes.

A

Direction cosines: $\cos\alpha = \frac{3}{5\sqrt{2}}$, $\cos\beta = \frac{4}{5\sqrt{2}}$, $\cos\gamma = \frac{1}{\sqrt{2}}$; Components: $3\hat{i}$, $4\hat{j}$, $5\hat{k}$

B

Direction cosines: $\cos\alpha = \frac{3}{5}$, $\cos\beta = \frac{4}{5}$, $\cos\gamma = 1$; Components: $3\hat{i}$, $4\hat{j}$, $5\hat{k}$

C

Direction cosines: $\cos\alpha = \frac{3}{\sqrt{2}}$, $\cos\beta = \frac{4}{\sqrt{2}}$, $\cos\gamma = \frac{5}{\sqrt{2}}$; Components: $3\hat{i}$, $4\hat{j}$, $5\hat{k}$

D

Direction cosines: $\cos\alpha = \frac{3}{5\sqrt{2}}$, $\cos\beta = \frac{2}{5\sqrt{2}}$, $\cos\gamma = \frac{1}{\sqrt{2}}$; Components: $\hat{i}$, $\hat{j}$, $\hat{k}$

Question 38: Palette 38/53

DPT MCQ

Find the cross product $\vec{a} \times \vec{b}$ if $\vec{a} = 3\hat{i} + 4\hat{j}$ and $\vec{b} = 2\hat{i} + 5\hat{j}$.

A

$7\hat{k}$

B

$-7\hat{k}$

C

$23\hat{k}$

D

$15\hat{k}$

Question 39: Palette 39/53

DPT MCQ

Find the cross product $\vec{a} \times \vec{b}$ if $\vec{a} = 4\hat{i} + 7\hat{j}$ and $\vec{b} = 2\hat{i} + 3\hat{j}$.

A

$-2\hat{k}$

B

$2\hat{k}$

C

$-26\hat{k}$

D

$26\hat{k}$

Question 40: Palette 40/53

DPT MCQ

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

A

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

B

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

C

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

D

$\tan^{-1}\left(\frac{n^2 - 1}{n^2 + 1}\right)$

Question 41: Palette 41/53

DPT MCQ

If $\vec{A} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{B} = 2\hat{i} - \hat{j} + 4\hat{k}$, then the unit vector along $\vec{A} + \vec{B}$ is:

A

$\frac{3\hat{i} + 5\hat{k}}{\sqrt{24}}$

B

$\frac{3\hat{i} - 5\hat{k}}{\sqrt{34}}$

C

$\frac{3\hat{i} + 5\hat{k}}{\sqrt{34}}$

D

None of these

Question 42: Palette 42/53

DPT MCQ

If $\vec{A} = 4\hat{i} + 3\hat{j}$ and $\vec{B} = 3\hat{i} + 4\hat{j}$, then the cosine of the angle between $\vec{A}$ and $\vec{A} + \vec{B}$ is:

A

$\frac{9\sqrt{2}}{5}$

B

$\frac{7}{5\sqrt{2}}$

C

$\frac{5\sqrt{2}}{49}$

D

$\frac{5\sqrt{2}}{28}$

Question 43: Palette 43/53

DPT MCQ

Given two vectors $\vec{A} = \hat{i} + \hat{j}$ and $\vec{B} = \hat{i} - \hat{j}$, match the terms in Column-I with Column-II:
Column-I:
(A) $(\vec{A} + \vec{B}) / 2$
(B) $(\vec{A} - \vec{B}) / 2$
(C) $(\vec{A} \cdot \vec{B}) / 2$
(D) $(\vec{A} \times \vec{B}) / 2$
Column-II:
(1) $\hat{i}$
(2) $\hat{j}$
(3) $-\hat{k}$
(4) $0$

A

(A) to (4); (B) to (1); (C) to (2); (D) to (2)

B

(A) to (2); (B) to (4); (C) to (3); (D) to (1)

C

(A) to (3); (B) to (2); (C) to (4); (D) to (1)

D

(A) to (1); (B) to (2); (C) to (4); (D) to (3)

Question 44: Palette 44/53

DPT MCQ

The component of vector $\vec{a} = 2\hat{i} + 3\hat{j}$ along the vector $\hat{i} + \hat{j}$ is

A

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

B

$10\sqrt{2}$

C

$5\sqrt{2}$

D

5

Question 45: Palette 45/53

DPT MCQ

What is the area of triangle formed by $\vec{A} = 2\hat{i} - 3\hat{j} + 4\hat{k}$ and $\vec{B} = \hat{i} - \hat{k}$ and their resultant?

A

$\sqrt{13.5}$ units

B

$13.5$ units

C

$\sqrt{38.7}$ units

D

$38.7$ units

Question 46: Palette 46/53

DPT MCQ

If $\vec{A} = 4\hat{i} + 6\hat{j}$ and $\vec{B} = 2\hat{i} + 3\hat{j}$, then which of the following is correct?

A

$\vec{A} \cdot \vec{B} = 29$

B

$\vec{A} \times \vec{B} = 0$

C

$\frac{\vert{}\vec{B}\vert{}}{\vert{}\vec{A}\vert{}} = \frac{2}{1}$

D

angle between $\vec{A}$ and $\vec{B}$ is $30^\circ$

Question 47: Palette 47/53

DPT MCQ

If $\vec{A} = 4\hat{i} + 6\hat{j}$ and $\vec{B} = 2\hat{i} + 3\hat{j}$, then which of the following is correct?

A

$\vec{A} \cdot \vec{B} = 29$

B

$\vec{A} \times \vec{B} = 0$

C

$\frac{\vert{}\vec{B}\vert{}}{\vert{}\vec{A}\vert{}} = \frac{2}{1}$

D

angle between $\vec{A}$ and $\vec{B}$ is $30^\circ$

Question 48: Palette 48/53

DPT MCQ

The vector $\vec{a} = \alpha\hat{i} + 2\hat{j} + \beta\hat{k}$ lies in the plane of the vectors $\vec{b} = \hat{i} + \hat{j}$ and $\vec{c} = \hat{j} + \hat{k}$ and bisects the angle between $\vec{b}$ and $\vec{c}$. Then which one of the following gives possible values of $\alpha$ and $\beta$?

A

$\alpha = 2, \beta = 2$

B

$\alpha = 1, \beta = 2$

C

$\alpha = 2, \beta = 1$

D

$\alpha = 1, \beta = 1$

Question 49: Palette 49/53

DPT MCQ

A vector $\vec{X}$, when added to two vectors $\vec{A} = 3\hat{i} - 5\hat{j} + 7\hat{k}$ and $\vec{B} = 2\hat{i} + 4\hat{j} - 3\hat{k}$ gives a unit vector along Y-axis. Find the vector $\vec{X}$.

A

$-5\hat{i} + 2\hat{j} - 4\hat{k}$

B

$5\hat{i} - 2\hat{j} + 4\hat{k}$

C

$-5\hat{i} - 2\hat{j} + 4\hat{k}$

D

$5\hat{i} + 2\hat{j} - 4\hat{k}$

Question 50: Palette 50/53

DPT MCQ

Two vectors of magnitude 10 N each gives a resultant of magnitude $10\sqrt{3}$ N. Find the angle between both vectors.

A

30°

B

45°

C

60°

D

90°

Question 51: Palette 51/53

DPT MCQ

Given that $\vec{A} + 2\vec{B} = x_1\hat{i} + y_1\hat{j}$ and $2\vec{A} - \vec{B} = x_2\hat{i} + y_2\hat{j}$, what is $\vec{A}$?

A

$\vec{A} = \frac{1}{5}(x_1 + 2x_2)\hat{i} + \frac{1}{5}(y_1 + 2y_2)\hat{j}$

B

$\vec{A} = \frac{1}{5}(x_1 + 2x_2)\hat{i} - \frac{1}{5}(y_1 + 2y_2)\hat{j}$

C

$\vec{A} = \frac{1}{5}(x_1 + 4x_2)\hat{i} + \frac{1}{5}(y_1 + 2y_2)\hat{j}$

D

$\vec{A} = \frac{1}{5}(x_1 + 4x_2)\hat{i} + \frac{1}{5}(y_1 + 4y_2)\hat{j}$

Question 52: Palette 52/53

DPT MCQ

A vector $\vec{A}$ is rotated by a small angle $\Delta\theta$ radians ($\Delta\theta \ll 1$) to get a new vector $\vec{B}$. In that case $\vert{}\vec{B} - \vec{A}\vert{}$ is

A

$0$

B

$\vert{}\vec{A}\vert{} \left(1 - \frac{\Delta\theta^2}{2}\right)$

C

$\vert{}\vec{A}\vert{} \Delta\theta$

D

$\vert{}\vec{B}\vert{} \Delta\theta - \vert{}\vec{A}\vert{}$

Question 53: Palette 53/53

DPT MCQ

A vector $\vec{A}$ is rotated by a small angle $\Delta\theta$ radians ($\Delta\theta \ll 1$) to get a new vector $\vec{B}$. In that case $\vert{}\vec{B} - \vec{A}\vert{}$ is

A

$0$

B

$\vert{}\vec{A}\vert{} \left(1 - \frac{\Delta\theta^2}{2}\right)$

C

$\vert{}\vec{A}\vert{} \Delta\theta$

D

$\vert{}\vec{B}\vert{} \Delta\theta - \vert{}\vec{A}\vert{}$