The instantaneous velocity at t=$\frac{\mathrm{\pi }}{2}$ of a particle whose positional equation is given by $x\left(t\right)=12 {\cos }^2\left(t\right)$ is $\left(v_{inst}=\frac{dx}{dt}\right)$ -

(1) 0

(2) -24

(3) 24

(4) $-12\sqrt{2}$

If acceleration of a particle is given as a(t) = sin(t)+2t. Then the velocity of the particle will be:
(acceleration $\mathrm{a}=\frac{\mathrm{dv}}{\mathrm{dt}}$)
1. \(-\cos(t)+ \frac{t^2}{2}\)
2. \(-\sin(t)+ t^2\)
3. \(-\cos(t)+ t^2\)
4. None of these

If \(x= 3\tan(t)\) and \(y = \sec (t)\), then the value of \(\frac{d^{2} y}{d x^{2}}~\text{at}~t = \frac{\pi}{4}\) is:
1. \(3\)
2. \(\frac{1}{18\sqrt{2}}\)
3. \(1\)
4. \(\frac{1}{6}\)

A particle's position as a function of time is given by $\mathrm{x}=-{\mathrm{t}}^2+6\mathrm{t}+3$. The maximum value of the position co-ordinate of the particle is:
1. \(8\)

2. \(12\)

3. \(3\)

4. \(6\)

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