A huge circular arc of length \(4.4\) \(ly\) subtends an angle \('4 {s}'\) at the centre of the circle. How long it would take for a body to complete \(4\) revolution if its speed is \(8 \;AU\;per\, second \;?\) Given : \(1\, {ly}=9.46 \times 10^{15} \,{m},\) \(\, {AU}=1.5 \times 10^{11}\, {m}\)

A proton and a deutron ( \(\mathrm{q}=+\mathrm{e}, m=2.0 \mathrm{u})\) having same kinetic energies enter a region of uniform magnetic field \(\vec{B}\), moving perpendicular to \(\vec{B}\). The ratio of the radius \(r_d\) of deutron path to the radius \(r_p\) of the proton path is _______.

The energy density associated with electric field \(\overrightarrow{ E }\) and magnetic field \(B\) of an electromagnetic wave in free space is given by ( \(\epsilon_0-\) permittivity of free space, \(\mu_0\) - permeability of free space)

An ideal monoatomic gas is confined in a cylinder by a spring loaded piston of cross section \(8.0\times10^{-3}\, m^2\) . Initially the gas is at \(300\, K\) and occupies a volume of \(2.4\times10^{-3}\, m^3\) and the spring is in its relaxed state as shown in figure. The gas is heated by a small heater until the piston moves out slowly by \(0.1\, m\). The force constant of the spring is \(8000\, N/m\) and the atmospheric pressure is \(1.0\times10^5\, N/m^2\) . The cylinder and the piston are thermally insulated. The piston and the spring are massless and there is no friction between the piston and the cylinder. The final temperature of the gas will be: (Neglect the heat loss through the lead wires of the heater . The heat capacity of the heater coil is also negligible)

Resistance of the wire is measured as \(2\,\Omega\) and \(3\,\Omega\) at \(10^{\circ}C\) and \(30^{\circ}C\) respectively. Temperature cocoefficient of resistance of the material of the wire is............\(^{\circ}C ^{-1}\)

A proton is moving undeflected in a region of crossed electric and magnetic fields at a constant speed of \(2 \times 10^5 \mathrm{~ms}^{-1}\). When the electric field is switched off, the proton moves along a circular path of radius 2 cm . The magnitude of electric field is \(x \times 10^4 \mathrm{~N} / \mathrm{C}\). The value of \(x\) is _______ Take the mass of the proton \(=1.6 \times 10^{-27} \mathrm{~kg}\).

An EM wave from air enters a medium. The electric fields are \(\overrightarrow {{E_1}}  = {E_{01}}\hat x\;cos\left[ {2\pi v\left( {\frac{z}{c} - t} \right)} \right]\) in air and \(\overrightarrow {{E_2}}  = {E_{02}}\hat x\;cos\left[ {k\left( {2z - ct} \right)} \right]\) in medium, where the wave number \(k\) and frequency \(v\) refer to their values in air. The medium is nonmagnetic. If \(\varepsilon {_{{r_1}}}\) and \(\varepsilon {_{{r_2}}}\) refer to relative permittivities of air and medium respectively, which of the following options is correct?

A student score the following marks in five tests : \(45, 54, 41, 57, 43\). His score is not known for the sixth test. If the mean score is \(48\) in the six tests, then the standard deviation of the marks in six tests is:

Let one end of a focal chord of the parabola \( y^{2}=16x \) be (16, 16). If \( P(\alpha,\beta) \) divides this focal chord internally in the ratio 5 : 2 then the minimum value of \( \alpha+\beta \) is equal to :

The oscillating magnetic field in a plane electromagnetic wave is given by \(B _{ y }=5 \times 10^{-6} \sin\) \(1000\,\pi\left(5 x -4 \times 10^{8} t \right) T\). The amplitude of electric field will be.

In an ac circuit, the instantaneous current is zero, when the instantaneous voltage is maximum. In this case, the source may be connected to: \(A\). pure inductor. \(B\). pure capacitor. \(C\). pure resistor. \(D\). combination of an inductor and capacitor. Choose the correct answer from the options given below:

The value of \(\cot \frac{\pi}{24}\) is :

A bullet of \(10\, {g}\), moving with velocity \(v\), collides head-on with the stationary bob of a pendulum and recoils with velocity \(100 \, {m} / {s}\). The length of the pendulum is \(0.5\, {m}\) and mass of the bob is \(1\, {kg}\). The minimum value of \(v=\) \(....{m} / {s}\) so that the pendulum describes a circle. (Assume the string to be inextensible and \({g}=10\, {m} / {s}^{2}\) )