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. $-\cos \left(\mathrm{t}\right)+\frac{{\mathrm{t}}^2}{2}$

2. $-\sin \left(\mathrm{t}\right)+{\mathrm{t}}^2$

3. $-\cos \left(\mathrm{t}\right)+{\mathrm{t}}^2$

4. None of these

If x=3tan(t) and y=sec(t), then the value of $\frac{d^{2}y}{d{x}^2} at t=\frac{\mathrm{\pi }}{4}$ is:

1. 3

2. $\frac{1}{18\sqrt{2}}$

3. 1

4. 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

The equation of position (x) of a particle is given by $x=(-3t^3+18t^2+5)m$. The maximum velocity of the particle is, if velocity is defined as $v=\frac{dx}{dt}$ :

1. +56 m/s

2. +46 m/s

3. +36 m/s

4. +26 m/s

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 sec

2. t=2 sec

3. t= 6 sec

4. t=3 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 $F\left(x\right)=(x^2-2x+1)N$, 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=$\int \mathrm{Fdt}$. 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.  $\left(\overrightarrow{\mathrm{A}} + \overrightarrow{\mathrm{B}}\right)·\left(\overrightarrow{\mathrm{A}} - \overrightarrow{\mathrm{B}}\right) \ne 0$