A radioactive nucleus can decay by two different processes. The half lives of the first and second decay processes are \(5 \times 10^3\) and \(10^5\) years respectively. Then, the effective half-life of the nucleus is

The magnitude of maximum acceleration is \(\pi\) times that of maximum velocity of a simple harmonic oscillator. The time period of the oscillator in seconds is

An electron is released from a distance of $4 \mathrm{m}$ from a stationary point charge $20 \mathrm{nC}$. What will be the speed of the electron when it is $2 \mathrm{m}$ away from the point charge?
[Charge of electron $=1.6\times {10}^{-19}\mathrm{C}$, mass of electron $=9\times {10}^{-31} \mathrm{kg}$, $\frac{1}{4\pi {\epsilon }_0}=9\times {10}^9$ S.I. unit]
 

An ideal gas is placed in a tank at $27 °\mathrm{C}$. The pressure is initially $600 \mathrm{kPa}$. One fourth of the gas is then released from the tank and thermal equilibrium is established. What will be the pressure if the temperature is $327 °\mathrm{C}$?

If the bulk modulus of water is \(2 \times 10^9 \mathrm{Nm}^{-2}\), then the required pressure to reduce the given volume of water by \(2 \%\) is

One end each of a resistance \(r\) capacitor \(C\) and resistance \(2 r\) are connected together. The other ends are respectively connected to the positive terminals of batteries, \(P, Q, R\) having respectively emf's \(E, E\) and \(2 E\). The negative terminals of the batteries are then connected together. In this circuit, with steady current the potential drop across the capacitor is :

The displacement equations of sound waves produced by two sources are given by \(y_1=5 \sin 400 \pi t\) and \(y_2=\) \(8 \sin 408 \pi t\), where ' \(t\) ' is time in seconds. If the waves are produced simultaneously, the number of beats produced per minute is

A vessel contains hydrogen and nitrogen gases in the ratio \(2: 3\) by mass. If the temperature of the mixture of the gases is \(30^{\circ} \mathrm{C}\), then the ratio of the average kinetic energies per molecule of hydrogen and nitrogen gases is (Molecular mass of hydrogen \(=2\) and molecular mass of nitrogen \(=28\) )

The radius of \({ }_{72} \mathrm{Te}^{125}\) nucleus is 6 fermi. The radius of \({ }_{13} \mathrm{Al}^{27}\) nucleus in meters is

The refractive index of the material of a small angled prism is 1.6 . If the angle of minimum deviation is \(4.2^{\circ}\), the angle of the prism is

Match the following table. \begin{array}{|c|c|c|c|} \hline & List-I & & List-II \ \hline A. & \begin{array}{l}\text { Michelson-Morley } \ \text { experiment }\end{array} & I. & \begin{array}{l}\text { The existence of } \ \text { anti-matter }\end{array} \ \hline B. & \begin{array}{l}\text { Stern-Gerlach } \ \text { experiment }\end{array} & II. & \begin{array}{l}\text { The existence of } \ \text { de-Broglie matter waves }\end{array} \ \hline C. & \begin{array}{l}\text { Davisson-Germer } \ \text { experiment }\end{array} & III. & Electrons have spins \ \hline D. & \begin{array}{l}\text { Anderson discovery } \ \text { of positron }\end{array} & \mathrm{IV}. & \begin{array}{l}\text { The non-existence of } \ \text { ether }\end{array} \ \hline \end{array} \(\begin{array}{llll}A & B & C & D\end{array}\)

Which of the following statement is incorrect?

An alternating voltage \(\varepsilon=30 \sin 200 t\) (in volts) is applied to the circuit below. The amplitude of the current through the circuit is