A gas undergoes a process where pressure P=$3V^4$, at every instant, here V is the volume. Find an expression for the bulk modulus (B) in terms of volume if it is related to P and V as $B=-\frac{VdP}{dV}$.

(1) $-3V^3$

(2) $-12 V^3$

(3) $-12 V^4$

(4) $-12 V^5$

The current in a circuit is defined as $I=\frac{dq}{dt}$. The charge (q) flowing through a circuit, as a function of time (t), is given by $q=5t^2-20t+3$. The minimum charge flows through the circuit at:
1. \(t = 4~\text{s}\)

2. \(t = 2~\text{s}\)

3. \(t = 6~\text{s}\)

4. \(t = 3~\text{s}\)

The current in a circuit is defined as $I=\frac{dq}{dt}$. If the charge (q) flowing through a circuit is given by $q=(t^2-3t+4),$ current flowing through the circuit is equal to zero at:

1. t=3 sec

2. t=2 sec

3. t=1.5 sec

4. t=15 sec

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

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

The angle made by the vector $\overrightarrow{\mathrm{A}} = \sqrt{3}\hat{\mathrm{i}} + 3\hat{\mathrm{j}} + 2\hat{\mathrm{k}}$ with y-axis is:

1.  ${\sin }^{-1}\left(\frac{3}{\sqrt{14}}\right)$

2.  ${\sin }^{-1}\left(\frac{\sqrt{7}}{4}\right)$

3.  ${\cos }^{-1}\left(\frac{4}{3}\right)$

4.  ${\cos }^{-1}\left(\frac{3}{5}\right)$

In the space, if the sum of vectors of unequal magnitude is zero, then the minimum number of vectors are:
1. \(2\)
2. \(3\)
3. \(4\)
4. \(5\)

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