If v(t) = 3t-1 and x(2) = 1, then the original position function is: 
Hint: \(\left(v \left( t \right) = \frac{d s}{d t}\right)\)$$
1. $\frac{3}{2}{t}^2-t-3$

2. $\frac{1}{2}{t}^2-t-3$

3. $\frac{3}{2}{t}^2-2t-3$

4. None of the above

If charge flown through a wire is given by q=3sin(3t), then-current flown through the wire at $\mathrm{t}=\frac{\mathrm{\pi }}{9}$ seconds is: $\left(\mathrm{i}=\frac{\mathrm{dq}}{\mathrm{dt}}\right)$

1. 4.5 Amp

2. 4.5$\sqrt{3}$ Amp

3. $\frac{\sqrt{3}}{2}$ Amp

4. 9 Amp

A weight hanging from a spring is stretched down 3 cm beyond its rest position and released at time t=0 to bob up and down. Its position at any later time t is s=3cos(t). Then its velocity at time t is $\left(velocity =\frac{ds}{dt}\right)$

(1) cost

(2) 3cost

(3) 3sint

(4) -3sint

The position of a particle is given by \(s\left( t\right) = \frac{2 t^{2} + 1}{t + 1}\). Then, at \(t= 2\), its velocity is: \(\left(v_{inst}= \frac{ds}{dt}\right)\)
1. \(\frac{16}{3}\)
2. \(\frac{15}{9}\)
3. \(\frac{15}{3}\)
4. None of these

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