Work done by a force (\(F\)) in displacing a body by dx is given by W=$\int F\left(x\right).dx$. If the force is given as a function of displacement (\(x\)) by \(F \left(x\right) = \left( x^{2} - 2 x + 1\right) \text{N}\), then work done by the force from \(x=0\) to \(x=3\) m is:

1. \(3\) J

2. \(6\) J

3. \(9\) J

4. \(21\) J

The impulse due to a force on a body is given by \(I=\int Fdt\). If the force applied on a body is given as a function of time \((t)\) as \(F = \left(3 t^{2} + 2 t + 5\right) \text{N}\), then impulse on the body between \(t = 3~\text{s}\) to \(t =5~\text{s}\) 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 $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}$, then which of the following is the unit vector perpendicular to $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}$?

1. $\frac{\hat{\mathrm{A}}<mspace\; width="1em"></mspace>\times <mspace\; width="1em"></mspace>\hat{\mathrm{B}}}{\mathrm{AB}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\mathrm{\theta }}$

2. $\frac{\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>\times <mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}}{\mathrm{AB}<mspace\; width="1em"></mspace>\cos <mspace\; width="1em"></mspace>\mathrm{\theta }}$

3. $\frac{\overrightarrow{\mathrm{A}}\times \overrightarrow{\mathrm{B}}}{\mathrm{AB}<mspace\; width="1em"></mspace>\sin <mspace\; width="1em"></mspace>\mathrm{\theta }}$

4. $\frac{\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>\times <mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}}{\mathrm{AB}<mspace\; width="1em"></mspace>}$

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

2. $\frac{\mathrm{A}}{\mathrm{B}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\frac{1}{2}$

3. $\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{B}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>48$

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

If $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}$ is perpendicular to $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}$  , then which of the following statement is correct?

1. $\left|\overrightarrow{\mathrm{A}}\right|<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\left|\overrightarrow{\mathrm{B}}\right|$

2. $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>\perp <mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}$

3. $\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{B}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\mathrm{zero}$

4. $\left(\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}\right)·\left(\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>-<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}\right)<mspace\; width="1em"></mspace>\ne <mspace\; width="1em"></mspace>0$

The angle between the two vectors \(\left(- 2 \hat{i} +3 \hat{j} + \hat{k}\right)\) and \(\left(\hat{i} + 2 \hat{j} - 4 \hat{k}\right)\) is:
1. \(0^{\circ}\)
2. \(90^{\circ}\)
3. \(180^{\circ}\)
4. \(45^{\circ}\)

If a unit vector \(\hat j\) is rotated through an angle of \(45^{\circ}\) anticlockwise, then the new vector will be:
1. \(\sqrt{2}\hat i + \sqrt{2}\hat j\)
2. \(\hat i + \hat j\)
3. \(\frac{1}{\sqrt{2}}\hat i + \frac{1}{\sqrt{2}}\hat j\)
4. \(-\frac{1}{\sqrt{2}}\hat i + \frac{1}{\sqrt{2}}\hat j\)

If $\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>2\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\hat{\mathrm{j}}$ and $\overrightarrow{\mathrm{b}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>3\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>2\hat{\mathrm{j}}$, then $\left|\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>\times <mspace\; width="1em"></mspace>\overrightarrow{\mathrm{b}}\right|=?$ 

\(\overrightarrow A\) and \(\overrightarrow {B}\) are two vectors given by \(\overrightarrow {A}= 2\hat i + 3\hat j\) and \(\overrightarrow {B}= \hat i + \hat j\). The component of \(\overrightarrow A\) parallel to \(\overrightarrow B\) is:
1. \(\frac{(2\hat i -\hat j)}{2}\)
2. \(\frac{5}{2}(\hat i - \hat j)\)
3. \(\frac{5}{2}(\hat i + \hat j)\)
4. \(\frac{(3\hat i -2\hat j)}{2}\)

If a vector is inclined at angles \(\alpha ,\beta ,~\text{and}~\gamma\), with \(x\), \(y\), and \(z\)-axis respectively, then the value of \(\sin^{2}\alpha+\sin^{2}\beta+ \sin^{2}\gamma\)${}^{}$
is equal to:
1. \(0\)
2. \(1\)
3. \(2\)
4. \(\frac{1}{2}\)