The impulse due to a force on a body is given by I=$\int \mathrm{Fdt}$. If the force applied on a body is given as a function of time (t) as $\mathrm{F}=(3{\mathrm{t}}^2+2\mathrm{t}+5)\mathrm{N}$, then impulse on the body between t=3 sec to t = 5 sec is:

1. 175 kg-m/sec

2. 41 kg-m/sec

3. 216 kg-m/sec

4. 124 kg-m/sec

If $\mathrm{\theta }$ is the angle between vectors $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$, then which of the following is the unit vector perpendicular to $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$?

1.  $\frac{\hat{\mathrm{A}} \times \hat{\mathrm{B}}}{\mathrm{AB} \sin \mathrm{\theta }}$

2.  $\frac{\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}}{\mathrm{AB} \cos \mathrm{\theta }}$

3.  $\frac{\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}}{\mathrm{AB} \sin \mathrm{\theta }}$

4.  $\frac{\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}}}{\mathrm{AB} }$

Which of the following option is not true, if $\overrightarrow{\mathrm{A}} = 3\hat{\mathrm{i}} + 4\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}} = 6\hat{\mathrm{i}} + 8\hat{\mathrm{j}}$, where \(\mathrm{A}\) and \(\mathrm{B}\) are the magnitudes of $\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$?
1. $\overrightarrow{\mathrm{A}} \times \overrightarrow{\mathrm{B}} = \overrightarrow{0}$

2. $\frac{\mathrm{A}}{\mathrm{B}} = \frac{1}{2}$

3. $\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{B}} = 48$

4. \(\mathrm{A}=5\)

If $\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}$ is perpendicular to $\overrightarrow{\mathrm{A}} - \overrightarrow{\mathrm{B}}$  , then which of the following statement is correct?

1.  $\left|\overrightarrow{\mathrm{A}}\right| = \left|\overrightarrow{\mathrm{B}}\right|$

2.  $\overrightarrow{\mathrm{A}} \perp \overrightarrow{\mathrm{B}}$

3.  $\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{B}} = \mathrm{zero}$

4.  $\left(\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}\right)·\left(\overrightarrow{\mathrm{A}} - \overrightarrow{\mathrm{B}}\right) \ne 0$

The angle between the two vectors $\left(-2\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + \hat{\mathrm{k}}\right)$ and $\left(\hat{\mathrm{i}} + 2\hat{\mathrm{j}} - 4\hat{\mathrm{k}}\right)$ is:

1. 0°

2. 90°

3. 180°

4. 45°

If a unit vector $\hat{j}$ is rotated through an angle of 45° anticlockwise, then the new vector will be:

1.  $\sqrt{2}\hat{\mathrm{i}} + \sqrt{2}\hat{\mathrm{j}}$

2.  $\hat{\mathrm{i}} + \hat{\mathrm{j}}$

3.  $\frac{1}{\sqrt{2}}\hat{\mathrm{i}} + \frac{1}{\sqrt{2}}\hat{\mathrm{j}}$

4.  $-\frac{1}{\sqrt{2}}\hat{\mathrm{i}} + \frac{1}{\sqrt{2}}\hat{\mathrm{j}}$

If $\overrightarrow{\mathrm{a}} = 2\hat{\mathrm{i}} + \hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{b}} = 3\hat{\mathrm{i}} + 2\hat{\mathrm{j}}$, then $\left|\overrightarrow{\mathrm{a}} \times \overrightarrow{\mathrm{b}}\right|=?$ 

1.  1

2.  $\sqrt{65}$

3.  8

4.  4 

$\overrightarrow{\mathrm{A}} \mathrm{and} \overrightarrow{\mathrm{B}}$ are two vectors given by $\overrightarrow{\mathrm{A} }= 2\hat{\mathrm{i}} + 3\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{B}} = \hat{\mathrm{i}} + \hat{\mathrm{j}}$. The component of $\stackrel{\to \; }{\mathrm{A}}$parallel to $\overrightarrow{\mathrm{B}}$ is:

1.  $\frac{1}{2}\left(2\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$

2.  $\frac{5}{2}\left(\hat{\mathrm{i}} - \hat{\mathrm{j}}\right)$

3.  $\frac{5}{2}\left(\hat{\mathrm{i}} + \hat{\mathrm{j}}\right)$

4.  $\frac{\left(3\hat{\mathrm{i}} - 2\hat{\mathrm{j}}\right)}{2}$

If a vector is inclined at angles, $\mathrm{\alpha }, \mathrm{\beta }, \mathrm{and} \mathrm{\gamma }$ with x, y, and z-axis respectively, then the value of ${\sin }^2 \mathrm{\alpha } + {\sin }^2 \mathrm{\beta }+ {\sin }^2 \mathrm{\gamma }$ is equal to

1. 0

2. 1

3. 2

4. $\frac{1}{2}$

A force of \(20\) N acts on a particle along a direction, making an angle of \(60^\circ\) with the vertical. The component of the force along the vertical direction will be: