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 given by: \(I=\dfrac{dq}{dt}.\) If the charge flowing through the circuit is described by: \(q=(t^2-3t+4),\) at what time is the current in the circuit equal to zero?
1. \(t=3\) s
2. \(t=2\) s
3. \(t=1.5\) s
4. \(t=15\) s

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 $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>3\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>4\hat{\mathrm{j}}$ 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. \(\mathrm{A}=5\)

The angle made by the vector $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\sqrt{3}\hat{\mathrm{i}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>3\hat{\mathrm{j}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>2\hat{\mathrm{k}}$ 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 $\overrightarrow{\mathrm{A}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{B}}$ 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 $\overrightarrow{\mathrm{a}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{b}}<mspace\; width="1em"></mspace>+<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{c}}<mspace\; width="1em"></mspace>=<mspace\; width="1em"></mspace>\overrightarrow{0}$; then which of the following statements is incorrect?

(1) $\overrightarrow{\mathrm{a}},<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{b}}<mspace\; width="1em"></mspace>\mathrm{and}<mspace\; width="1em"></mspace>\overrightarrow{\mathrm{c}}$ must each be a null vector.

(2) The magnitude of $\overrightarrow{\mathrm{a}}+\overrightarrow{\mathrm{b}}$ equals the magnitude of $\overrightarrow{\mathrm{c}}$.

(3) The magnitude of ä can never be greater than the sum of the magnitudes of $\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$

(4) ä must lie in the plane of $\overrightarrow{\mathrm{b}}$ and $\overrightarrow{\mathrm{c}}$.