A particle with restoring force proportional to displacement and resisting force proportional to velocity is subjected to a force $\mathrm{Fsin\omega t}$ . If the amplitude of the particle is maximum for $\mathrm{\omega }={\mathrm{\omega }}_1$  and the energy of the particle is maximum for $\mathrm{\omega }={\mathrm{\omega }}_2$, then (where ${\omega }_0$ is natural frequency of oscillation of particle)

1. ${\mathrm{\omega }}_1={\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2\ne {\mathrm{\omega }}_0$

2. ${\mathrm{\omega }}_1={\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2={\mathrm{\omega }}_0$

3. ${\mathrm{\omega }}_1\ne {\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2={\mathrm{\omega }}_0$

4. ${\mathrm{\omega }}_1\ne {\mathrm{\omega }}_0$ and ${\mathrm{\omega }}_2\ne {\mathrm{\omega }}_0$

The amplitude of velocity of a particle is given by, ${\mathrm{V}}_{\mathrm{m}}={\mathrm{V}}_0/\left({\mathrm{a\omega }}^2-\mathrm{b\omega }+\mathrm{c}\right)$ where ${\mathrm{V}}_0,\mathrm{a},\mathrm{b},\mathrm{c}$ are positive:

The condition for a single resonant frequency is 

1. ${\mathrm{b}}^2<4\mathrm{ac}$

2. ${\mathrm{b}}^2=4\mathrm{ac}$

3. ${\mathrm{b}}^2=5\mathrm{ac}$

4. ${\mathrm{b}}^2=7\mathrm{ac}$

In a forced oscillation, when the system oscillates under the action of the driving force $\mathrm{F}$ $=$ ${\mathrm{F}}_0$ $\sin$ $\mathrm{\omega t}$ in addition to its internal restoring force, the particle oscillates with a frequency equal to

1.  The natural frequency of the body

2.  Frequency of driving force

3.  The difference in frequency of driving force and natural frequency

4.  Mean of the driving frequency and natural frequency

In the following questions, a statement of assertion (A) is followed by a statement of the reason (R).

1.  If both Assertion & Reason are true and the reason is the correct explanation of the assertion, then mark (1).

2.  If both Assertion & Reason are true but the reason is not the correct explanation of the assertion, then mark (2).

3.  If Assertion is a true statement but Reason is false, then mark (3).

4.  if both Assertion and Reason are false statements, then mark (4).

A: The periodic time of a hard spring is more as compared to the soft spring.

R: The spring constant of hard spring is less.

For forced oscillations, a particle oscillates in a simple harmonic fashion with a frequency equal to:

1. the frequency of driving force.

2. the mean of frequency of driving force and natural frequency of the body.

3. the difference of frequency of driving force and natural frequency of the body.

4. the natural frequency of the body.

Resonance is a special case of:
1. forced oscillations
2. damped oscillations
3. undamped oscillations
4. coupled oscillations

The figure given below shows the graphs for amplitudes of forced oscillations in resonance conditions for different damping conditions. 

One of the conclusions that can be drawn from the graph above is:

1. As damping increases, amplitude increases

2. As damping increases, the amplitude decreases

3. As damping increases, the amplitude does not change

4. As damping increases, the amplitude may increase or decrease