The time period of revolution of electron in its ground state orbit in a hydrogen atom is \(1.6 \times 10^{-16} \;\mathrm{s}\). The frequency of revolution of the electron in its first excited state (in \(s^{-1}\) ) is:

A capacitor of capacitance \(C=1\, \mu \,{F}\) is suddenly connected to a battery of \(100\, volt\) through a resistance \({R}=100\, \Omega\). The time taken for the capacitor to be charged to get \(50 \,{V}\) is \(....\,\times \,10^{-4}\,s.\) [Take \(\ln 2=0.69]\)

A body starts falling freely from height \(\mathrm{H}\) hits an inclined plane in its path at height \(\mathrm{h}\). As a result of this perfectly elastic impact, the direction of the velocity of the body becomes horizontal. The value of \(\frac{\mathrm{H}}{\mathrm{h}}\) for which the body will take the maximum time to reach the ground is _______.

A series \(LCR\) circuit driven by \(300\, {V}\) at a frequency of \(50 \,{Hz}\) contains a resistance \({R}=3 \,{k} \Omega\), an inductor of inductive reactance \(X_{L}=250 \,\pi \Omega\) and an unknown capacitor. The value of capacitance to maximize the average power should be : (Take \(\pi^{2}=10\) ) (in \(\mu {F}\))

In a transverse wave the distance between a crest and neighbouring trough at the same instant is \(4.0\, cm\) and the distance between a crest and trough at the same place is \(1 .0\, cm\) . The next crest appears at the same place after a time interval of \(0.4\,s\). The maximum speed of the vibrating particles in the medium is

Given below are two statements. One is labelled as Assertion (A) and the other is labelled as Reason (R).
Assertion (A) : In an insulated container, a gas is adiabatically shrunk to half of its initial volume. The temperature of the gas decreases.
Reason (R): Free expansion of an ideal gas is an irreversible and an adiabatic process.
In the light of the above statements, choose the correct answer from the options given below :

Which of the following equations is dimensionally incorrect? Where \(t=\) time, \(h=\) height, \(s=\) surface tension, \(\theta=\) angle, \(\rho=\) density, \(a, r=\) radius, \(g=\) acceleration due to gravity, \({v}=\) volume, \({p}=\) pressure, \({W}=\) work done, \(\Gamma=\) torque, \(\varepsilon=\) permittivity, \({E}=\) electric field, \({J}=\) current density, \({L}=\) length.

A solution containing active cobalt \({}_{27}^{60}Co\) having activity of \(0.8\,\mu Ci\) and decay constant \(\lambda \) is injected in an animal's body. If \(1 \,cm^3\) of blood is drawn from the animal's body after \(10\, hrs\) of injection, the activity found was \(300\, decays\) per minute. What is the volume of blood that is flowing in the body?..........\(litres\) ( \(ICi = 3.7 \times 10^{10}\) decay per second and at \(t = 10\, hrs\) \({e^{ - \lambda t}} = 0.84\) )

In a long glass tube, mixture of two liquids A and B with refractive indices 1.3 and 1.4 respectively, forms a convex refractive meniscus towards \(A\). If an object placed at 13 cm from the vertex of the meniscus in A forms an image with a magnification of ' \(-2^{\prime}\) then the radius of curvature of meniscus is :

Young’s moduli of two wires \(A\) and \(B\) are in the ratio \(7 : 4\). Wire \(A\) is \(2\, m\) long and has radius \(R\). Wire \(A\) is \(2\, m\) long and has radius \(R\). Wire \(B\) is \(1.5\, m\) long and has radius \(2\, mm\). If the two wires stretch by the same length for a given load, then the value of \(R\) is close to ......... \(mm\)

Two point charges \(q_1\,(\sqrt {10}\,\,\mu C)\) and \(q_2\,(-25\,\,\mu C)\) are placed on the \(x-\) axis at \(x = 1\,m\) and \(x = 4\,m\) respectively. The electric field (in \(V/m\) ) at a point \(y = 3\,m\) on \(y-\) axis is, [ take \({\mkern 1mu} {\mkern 1mu} \frac{1}{{4\pi {\varepsilon _0}}} = 9 \times {10^9}{\mkern 1mu} {\mkern 1mu} N{m^2}{C^{ - 2}}{\rm{ }}\) ]