An ideal gas is enclosed in a cylinder at pressure of \(2\,atm\) and temperature, \(300\,K.\) The mean time between two successive collisions is \(6 \times 10^{-8}\, s.\) If the pressure is doubled and temperature is increased to \(500\,K,\) the mean time between two successive collisions will be close to

A string of length \(L\) is fixed at one end and carries a mass of \(M\) at the other end. The mass makes \(\left(\frac{3}{\pi}\right)\) rotations per second about the vertical axis passing through end of the string as shown. The tension in the string is \(\ldots\ldots\) ML.

For the \(LCR\) circuit, shown here, the current is observed to lead the applied voltage. An additional capacitor \(C '\) , when joined with the capacitor \(C\) present in the circuit , makes the power factor of the circuit unity. The capacitor \(C'\), must have been connected in

The decay of a proton to neutron is :

The velocity of a particle executing SHM varies with displacement \(( x )\) as \(4 v ^2=50- x ^2\). The time period of oscillations is \(\frac{x}{7} s\). The value of \(x\) is \(............\) \(\left(\right.\) Take \(\left.\pi=\frac{22}{7}\right)\)

If the length of the pendulum in pendulum clock increases by \(0.1\, \%\), then the error in time per day is: (in \(s\))

An \(emf\) of \(0.08\,V\) is induced in a metal rod of length \(10\,cm\) held normal to a uniform magnetic field of \(0.4\,T\), when moves with a velocity of ...........\(ms^{-1}\)

In an ammeter, \(5 \%\) of the main current passes through the galvanometer. If resistance of the galvanometer is G, the resistance of ammeter will be _______.

For the circuit shown above, equivalent GATE is :

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : The binding energy per nucleon is found to be practically independent of the atomic number A, for nuclei with mass numbers between 30 and 170.
Reason (R): Nuclear force is long range.
In the light of the above statements, choose the correct answer from the options given below :

The number density of molecules of a gas depends on their distance \(r\) from the origin as, \(n\left( r \right) = {n_0}{e^{ - \alpha {r^4}}}\). Then the total number of molecules is proportional to

A small bob of mass 100 mg and charge \(+10 \mu \mathrm{C}\) is connected to an insulating string of length 1 m. It is brought near to an infinitely long nonconducting sheet of charge density ' \(\sigma\) ' as shown in figure. If string subtends an angle of \(45^{\circ}\) with the sheet at equilibrium the charge density of sheet will be :
(Given, \(\varepsilon_0=8.85 \times 10^{-12} \frac{\mathrm{~F}}{\mathrm{~m}}\) and acceleration due to gravity, \(g=10 \mathrm{~m} / \mathrm{s}^2\))

Hot water cools from \(60\,^oC\) to \(50\,^oC\) in the first \(10\,minutes\) and to \(42\,^oC\) in the next \(10\,minutes.\) The temperature of the surroundings is ...... \(^oC\)