The de Broglie wavelength of an oxygen molecule at \(27^{\circ} C\) is \(x \times 10^{-12} m\). The value of \(x\) is (take Planck's constant \(=6.63 \times 10^{-34} J . s\), Boltzmann constant \(=1.38 \times 10^{-23} J / K\), mass of oxygen. Molecule \(\left.=5.31 \times 10^{-26} kg\right)\).

A cylinder with adiabatic walls is closed at both ends and is divided into two compartments by a frictionless adiabatic piston. Ideal gas is filled in both (left and right) the compartments at same \(P, V, T\). Heating is started from left side until pressure changes to \(\dfrac{27P}{8}\). If initial volume of each compartment was \(9\) litres then the final volume in right-hand side compartment is __________ litres. (for this ideal gas \(C_P/C_V = 1.5\))

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet \(S\) having surface charge density \(+\sigma\). The electron at \(t=0\) is at a distance of 1 m from \(S\) and has a speed of \(1 m s ^{-1}\). The maximum value of \(\sigma\), if the electron strikes \(S\) at \(t=1 s\) is \(\alpha\left[\frac{m \epsilon_0}{e}\right] \frac{ C }{ m ^2}\). The value of \(\alpha\) is __________ .

In finding the electric field using Gauss Law the formula \(|\overrightarrow{\mathrm{E}}|=\frac{q_{\mathrm{enc}}}{\varepsilon_{0}|\mathrm{A}|}\) is applicable. In the formula \(\varepsilon_{0}\) is permittivity of free space, \(A\) is the area of Gaussian surface and \(q_{enc}\) is charge enclosed by the Gaussian surface. The equation can be used in which of the following situation?

A body \(A,\) of mass \(m=0.1\; kg\) has an initial velocity of \(3 \hat{\mathrm{i}}\; \mathrm{ms}^{-1} .\) It collides elastically with another body, \(\mathrm{B}\) of the same mass which has an initial velocity of \(5 \hat{\mathrm{j}} \;\mathrm{ms}^{-1}\). After collision. A moves with a velocity \(\overline{\mathrm{v}}=4(\hat{\mathrm{i}}+\hat{\mathrm{j}})\). The energy of \(\mathrm{B}\) after collision is written as \(\frac{\mathrm{x}}{10} \;\mathrm{J}\) The value of \(x\) is

Initially a satellite of 100 kg is in a circular orbit of radius \(1.5 R _{ E }\). This satellite can be moved to a circular orbit of radius \(3 R _{ E }\) by supplying \(\alpha \times 10^6 J\) of energy. The value of \(\alpha\) is _________ .
(Take Radius of Earth \(R_E=6 \times 10^6 m\) and \(g =10 m / s ^2\) )

The energy released if hydrogen atoms are combined to form \(^{4}_{2}\text{He}\) is __________ MeV. (Take binding energies per nucleon of \(^{2}_{1}\text{H}\) and \(^{4}_{2}\text{He}\) as \(1.1\) MeV and \(7.2\) MeV, respectively)

Sun light falls normally on a surface of area \(36\,cm ^{2}\) and exerts an average force of \(7.2 \times 10^{-9}\,N\) within a time period of \(20\) minutes. Considering a case of complete absorption, the energy flux of incident light is.

Given below are two statements:
Statement I: For all elements, greater the mass of the nucleus, greater is the binding energy per nucleon.
Statement II: For all elements, nuclei with less binding energy per nucleon transforms to nuclei with greater binding energy per nucleon.
In the light of the above statements, choose the correct answer from the options given below:

Two resistance of \(100\ \Omega\) and \(200\  \Omega\) are connected in series with a battery of \(4 \mathrm{~V}\) and negligible internal resistance. \(A\) voltmeter is used to measure voltage across \(100 \Omega\) resistance, which gives reading as \(1 \mathrm{~V}\). The resistance of voltmeter must be _______ \(\Omega\).

The relative permittivity of distilled water is \(81 .\) The velocity of light in it will be \(....\,\times 10^{7}\,{m} / {s}\) \(\left(\right.\) Given \(\left.\mu_{r}=1\right)\)

When the temperature of a metal wire is increased from \(0^{\circ} \,C\) to \(10^{\circ}\, C\), its length increases by \(0.02 \% .\) The percentage change in its mass density will be closest to:

The position of a particle moving on \(x\)-axis is given by \(x(t)=A \sin t+B \cos ^2 t+C t^2+D\), where \(t\) is time. The dimension of \(\frac{A B C}{D}\) is