Two strings with circular cross section and made of same material, are stretched to have same amount of tension. A transverse wave is then made to pass through both the strings. The velocity of the wave in the first string having the radius of cross section R is \(\mathrm{v}_1\), and that in the other string having radius of cross section \(R / 2\) is \(v_2\). Then \(\frac{v_2}{v_1}=\) ________.

A force \(\mathrm{f}=\mathrm{x}^2 \mathrm{y} \hat{\mathrm{i}}+\mathrm{y}^2 \hat{\mathrm{j}}\) acts on a particle in a plane \(\mathrm{x}+\mathrm{y}=10\). The work done by this force during a displacement from \((0,0)\) to \((4 \mathrm{~m}, 2 \mathrm{~m})\) is Joule (round off to the nearest integer)

Primary coil of a transformer is connected to \(220 \mathrm{~V}\) ac. Primary and secondary turns of the transforms are \(100\) and \(10\) respectively. Secondary coil of transformer is connected to two series resistance shown in shown in figure. The output voltage \(\left(\mathrm{V}_0\right)\) is _______.

Given below are two statements: Statement \(I\) : In hydrogen atom, the frequency of radiation emitted when an electron jumps from lower energy orbit \(\left( E _{1}\right)\) to higher energy orbit \(\left( E _{2}\right)\), is given as \(hf = E _{1}- E _{2}\). Statement \(II\) : The jumping of electron from higher energy orbit \(\left(E_{2}\right)\) to lower energy orbit \(\left(E_{1}\right)\) is associated with frequency of radiation given as \(f\) \(=\left( E _{2}- E _{1}\right) / h\) This condition is Bohr's frequency condition. In the light of the above statements, choose the correct answer from the options given below

Consider two separate ideal gases of electrons and protons having same number of particles. The temperature of both the gases are same. The ratio of the uncertainty in determining the position of an electron to that of a proton is proportional to :-

Two coaxial discs, having moments of inertia \(I_1\) and \(\frac{I_1}{2}\) are a rotating with respectively angular velocities \(\omega_1\) and \(\frac{\omega_1}{2}\), about their common axes. They are brought in contact with each other and thereafter they rotate with a common angular velocity. If \(E_f\) and \(E_i\) are the final and initial total energies, then \((E_f -E_i)\) is

A body of mass \(5\, kg\) under the action of constant force \(\overrightarrow F  = {F_x}\hat i + {F_y}\hat j\) has velocity at \(t\,= 0\, s\) as \(\overrightarrow v  = \left( {6\hat i - 2\hat j\,m/s} \right)\)  and at \(t\, = 10\,s\) as \(\overrightarrow v  = +6\hat j\,m/s\). The force \(\overrightarrow F \) is 

Given below are two statements: Statement\(- I\) : Susceptibilities of paramagnetic and ferromagnetic substances increase with decrease in temperature. Statement\(- II\): Diamagnetism is a result of orbital motions of electrons developing magnetic moments opposite to the applied magnetic field. Choose the \(CORRECT\) answer from the options given below : -

Due to cold weather a \(1\, {m}\) water pipe of cross-sectional area \(1\, {cm}^{2}\) is filled with ice at \(-10^{\circ} {C}\). Resistive heating is used to melt the ice. Current of \(0.5\, {A}\) is passed through \(4\, {k} \Omega\) resistance. Assuming that all the heat produced is used for melting, what is the minimum time required ? (In \({s}\)) (Given latent heat of fusion for water/ice \(=3.33 \times 10^{5}\, {J} {kg}^{-1}\), specific heat of ice \(=2 \times 10^{3}\, {J}\) \({kg}^{-1}\) and density of ice \(=10^{3}\, {kg} / {m}^{3}\)

A concave mirror has radius of curvature of \(40\, cm\). It is at the bottom of a glass that has water filled up to \(5\, cm\) (see figure). If a small particle is floating on the surface of water, its image as seen, from directly above the glass, is at a distance \(d\) from the surface of water. The value of \(d\) is close to ......\(cm\) (Refractive index of water \(= 1.33\))

In a dilute gas at pressure \(P\) and temperature \(T\), the mean time between successive collisions of a molecule varies with \(T\) as