Activities of three radioactive substances \(A , B\) and \(C\) are represented by the curves \(A, B\) and \(C,\) in the figure. Then their half-lives \(T _{\frac{1}{2}}( A ): T _{\frac{1}{2}}( B ): T _{\frac{1}{2}}( C )\) are in the ratio

Magnetic flux (in weber) in a closed circuit of resistance \(20\,\Omega\) varies with time \(t(s)\) as \(\phi=8 t ^{2}-9 t +5\). The magnitude of the induced current at \(t =0.25\,s\) will be \(...mA\)

The electrostatic force \(\left(\vec{F}_1\right)\) and magnetic force \(\left(\vec{F}_2\right)\) acting on a charge \(q\) moving with velocity \(v\) can be written _______.

An alpha-particle of mass \(m\) suffers \(1-\) dimensional elastic collision with a nucleus at rest of unknown mass. It is scattered directly backwards losing, \(64\%\) of its initial kinetic energy. The mass of the nucleus is .......... \(\mathrm{m}\)

Three different processes that can occur in an ideal monoatomic gas are shown in the \(P\) vs \(V\) diagram. The paths are labelled as \(A \rightarrow B, A \rightarrow C\) and \(A \rightarrow D .\) The change in internal energies during these process are taken as \(E _{ AB }, E _{ AC }\) and \(E _{ AD }\) and the work done as \(W _{ AB }\) \(W _{ AC }\) and \(W _{ AD }\) The correct relation between these parameters are

Statement \(I:\) If three forces \(\vec{F}_{1}, \vec{F}_{2}\) and \(\vec{F}_{3}\) are represented by three sides of a triangle and \(\overrightarrow{{F}}_{1}+\overrightarrow{{F}}_{2}=-\overrightarrow{{F}}_{3}\), then these three forces are concurrent forces and satisfy the condition for equilibrium. Statement \(II:\) A triangle made up of three forces \(\overrightarrow{{F}}_{1}, \overrightarrow{{F}}_{2}\) and \(\overrightarrow{{F}}_{3}\) as its sides taken in the same order, satisfy the condition for translatory equilibrium. In the light of the above statements, choose the most appropriate answer from the options given below:

A time varying potential difference is applied between the plates of a parallel plate capacitor of capacitance \(2.5 \mu \mathrm{~F}\). The dielectric constant of the medium between the capacitor plates is 1 . It produces an instantaneous displacement current of 0.25 mA in the intervening space between the capacitor plates, the magnitude of the rate of change of the potential difference will be _____ \(\mathrm{Vs}^{-1}\).

A plano convex lens of refractive index \({\mu _1}\) and focal length \(f_1\) is kept in contact with another plano concave lens of refractive index \({\mu _2}\) and focal length \(f_2\). If the radius of curvature of their spherical faces is \(R\) each and \(f_1 = 2f_2\), then \({\mu _1}\) and \({\mu _2}\) are related as

The induced \(emf\) can be produced in a coil by \(A.\) moving the coil with uniform speed inside magnetic field \(B.\) moving the coil with non-uniform speed inside uniform magnetic field \(C.\) rotating the coil inside the uniform magnetic field \(D.\) changing the area of the coil inside the uniform magnetic field Choose the correct answer from the options given below:

Two gases-argon (atomic radius \(0.07 \;\mathrm{nm}\),atomic weight \(40\) ) and xenon (atomic radius \(0.1\; \mathrm{nm},\) atomic weight \(140\) ) have the same number density and are at the same temperature. The raito of their respective mean free times is closest to