Two particles that are initially at rest, move towards each other under the action of their mutual attraction. If their speeds are \(v\) and \(2v\) at any instant, then the speed of the centre of mass of the system will be:
1. \(2v\)
2. \(0\)
3. \(1.5v\)
4. \(v\)

A particle of mass \(M\) is situated at the centre of a spherical shell of the same mass and radius \(a.\) The gravitational potential at a point situated at \(a/2\) distance from the centre, will be:
1. \(-\dfrac{3GM}{a}\)
2. \(-\dfrac{2GM}{a}\)
3. \(-\dfrac{GM}{a}\)
4. \(-\dfrac{4GM}{a}\)

The device that can act as a complete electronic circuit is:

1. Junction diode

2. Integrated circuit

3. Junction transistor

4. Zener diode

A potentiometer circuit is set up as shown in the figure below. The potential gradient across the potentiometer wire is k volt/cm. Ammeter present in the circuit reads 1.0 A when the two-way key is switched off. The balance points, when the key between the terminals (i) 1 and 2 (ii) 1 and 3, is plugged in, are found to be at lengths $l_1$ $cm$ and $l_2$ $cm$ respectively. The magnitudes of the resistors R and X in ohm, are then, respectively, equal to: 
   

1. $k(l_2-l_1)$ $and$ $k{l}_2$

2. $k{l}_1$ $and$ $k(l_2-l_1)$

3. $k(l_2-l_1)$ $and$ $k{l}_1$

4. $k{l}_1$ $and$ $k{l}_2$

A tuning fork of frequency \(512~\text{Hz}\) makes \(4~\text{beats/s}\) with the vibrating string of a piano. The beat frequency decreases to \(2~\text{beats/s}\) when the tension in the piano string is slightly increased. The frequency of the piano string before increasing the tension was:
1. \(510~\text{Hz}\) 
2. \(514~\text{Hz}\) 
3. \(516~\text{Hz}\) 
4. \(508~\text{Hz}\) 

Six vectors $\overrightarrow{\mathrm{a}}$ through $\overrightarrow{\mathrm{f}}$ have the magnitudes and directions indicated in the figure. Which of the following statements is true? 

1. $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{f}}$

2. $\overrightarrow{\mathrm{d}}+\overrightarrow{\mathrm{c}}=\overrightarrow{\mathrm{f}}$

3. $\overrightarrow{\mathrm{d}}+\overrightarrow{\mathrm{e}}=\overrightarrow{\mathrm{f}}$

4. $\overrightarrow{\mathrm{b}}+\overrightarrow{\mathrm{e}}=\overrightarrow{\mathrm{f}}$

A gramophone record is revolving with an angular velocity $\omega$. A coin is placed at a distance r from the centre of the record. The static coefficient of friction is $\mu$. The coin will revolve with the record if:

1. $r=\mu g{\omega }^2$

2. $r<\frac{{\omega }^2}{\mu g}$

3. $r\le \frac{\mu g}{{\omega }^2}$

4. $r\ge \frac{\mu g}{{\omega }^2}$

Which of the following statement is false for the properties of electromagnetic waves?

The energy of a hydrogen atom in the ground state is \(-13.6\) eV. The energy of a \(\mathrm{He}^{+}\) ion in the first excited state will be:
1. \(-13.6\) eV
2. \(-27.2\) eV
3. \(-54.4\) eV
4. \(-6.8\) eV