One end of the string of length \(l\) is connected to a particle of mass \(m\) and the other end is connected to a small peg on a smooth horizontal table. If the particle moves in a circle with speed \(v\), the net force on the particle (directed towards the center) will be: (\(T\) represents the tension in the string)
1. \(T+\frac{m v^2}{l}\)
2. \(T-\frac{m v^2}{l}\)
3. zero
4. \(T\)

A particle executes linear simple harmonic motion with amplitude of \(3~\text{cm}\). When the particle is at \(2~\text{cm}\) from the mean position, the magnitude of its velocity is equal to that of its acceleration. Then its time period in seconds is:
1. \(\frac{\sqrt5}{2\pi}\)
2. \(\frac{4\pi}{\sqrt5}\)
3. \(\frac{4\pi}{\sqrt3}\)
4. \(\frac{\sqrt5}{\pi}\)

Two polaroids \(P_1\) and \(P_2\) are placed with their axis perpendicular to each other. Unpolarised light of intensity \(I_0\) is incident on \(P_1\). A third polaroid \(P_3\) is kept in between \(P_1\) and \(P_2\) such that its axis makes an angle \(45^\circ\) with that of \(P_1\). The intensity of transmitted light through \(P_2\) is:
1. \(\frac{I_0}{4}\)
2. \(\frac{I_0}{8}\)
3. \(\frac{I_0}{16}\)
4. \(\frac{I_0}{2}\)

The bulk modulus of a spherical object is \(B.\) If it is subjected to uniform pressure \(P,\) the fractional decrease in radius is:
1. \(\frac{B}{3P}\)
2. \(\frac{3P}{B}\)
3. \(\frac{P}{3B}\)
4. \(\frac{P}{B}\)

In an electromagnetic wave in free space, the root mean square value of the electric field is \(E_{\text{rms}} = 6~\text{V/m}\). The peak value of the magnetic field is:
1. \(2.83\times 10^{-8}~\text{T}\)
2. \(0.70\times 10^{-8}~\text{T}\)
3. \(4.23\times 10^{-8}~\text{T}\)
4. \(1.41\times 10^{-8}~\text{T}\)

A rope is wrapped around a hollow cylinder with a mass of \(3\) kg and a radius of \(40\) cm. What is the angular acceleration of the cylinder if the rope is pulled with a force of \(30\) N?
1. \(0.25 ~\text{rad/s}^2 \)
2. \(25 ~\text{rad/s}^2 \)
3. \(5 ~\text{m/s}^2 \)
4. \(25 ~\text{m/s}^2 \)

Two discs of the same moment of inertia are rotating about their regular axis passing through centre and perpendicular to the plane of the disc with angular velocities \(\omega_1\) ${\mathrm{}}_{}$and \(\omega_2\). They are brought into contact face to face with their axis of rotation coinciding. The expression for loss of energy during this process is:
1. \(\frac{1}{4}I\left(\omega_1-\omega_2\right)^2\)
2. \(I\left(\omega_1-\omega_2\right)^2\)
3. \(\frac{1}{8}I\left(\omega_1-\omega_2\right)^2\)
4. \(\frac{1}{2}I\left(\omega_1-\omega_2\right)^2\)

The photoelectric threshold wavelength of silver is \(3250\times 10^{-10}~\text{m}\). What will be the velocity of the electron ejected from a silver surface by the ultraviolet light of wavelength \(2536\times 10^{-10}~\text{m}\)? (Given \(h= 4.14\times 10^{-15}~\text{eVs}\)$$ and $\mathrm{}$\(c= 3\times 10^{8}~\text{m/s}\)${\mathrm{}}^{}$)
1. \(\approx 0.6\times 10^{6}~\text{m/s}\)
2. \(\approx 61\times 10^{3}~\text{m/s}\)
3. \(\approx 0.3\times 10^{6}~\text{m/s}\)
4. \(\approx 0.3\times 10^{5}~\text{m/s}\)

A \(250\) turn rectangular coil of length \(2.1\) cm and width \(1.25\) cm carries a current of \(85~\mu\text{A}\) and subjected to the magnetic field of strength \(0.85~\text{T}\). Work done for rotating the coil by \(180^\circ\) against the torque is:
1. \(4.55~\mu\text{J} \)
2. \(2.3~\mu\text{J} \)
3. \(1.15~\mu\text{J} \)
4. \(9.4~\mu\text{J} \)

The ratio of wavelengths of the last line of Balmer series and the last line of Lyman series is:
1. \(1\)
2. \(4\)
3. \(0.5\)
4. \(2\)