Identify the pair of physical quantities which have different dimensions 

The temperature of 1 mole of an ideal monoatomic gas is increased by \(50^{\circ} \mathrm{C}\) at constant pressure. The total heat added and change in internal energy are \(E_1\) and \(E_2\), respectively. If \(\frac{E_1}{E_2}=\frac{x}{9}\) then the value of \(x\) is _____

Find the gravitational force of attraction between the ring and sphere as shown in the diagram, where the plane of the ring is perpendicular to the line joining the centres. If \(\sqrt{8}\, R\) is the distance between the centres of a \(mng\) (of mass ' \(m\) ) and a sphere (mass ' \(M\) ) where both have equal radius \('R'.\)

Given below are two statements: one is labelled as Assertion \(A\) and the other is labelled as Reason \(R\). Assertion \(A\) : Body \('P'\) having mass \(M\) moving with speed \('u'\) has head-on collision elastically with another body \('Q'\) having mass \('m'\) initially at rest. If \(m< < M,\) body \('Q'\) will have a maximum speed equal to \('2u'\) after collision. Reason \(R\) : During elastic collision, the momentum and kinetic energy are both conserved. In the light of the above statements, choose the most appropriate answer from the options given below:

The velocity of a small ball of mass ' \(m\) ' and density \(d _{1}\), when dropped in a container filled with glycerine, becomes constant after some time. If the density of glycerine is \(d _{2}\), then the viscous force acting on the ball, will be

The value of current in the \(6 \,\Omega\) resistance is \(....\,A\)

A particle is projected with velocity \(v_{0}\) along \(x-\) axis. A damping force is acting on the particle which is proportional to the square of the distance from the origin i.e., \(ma =-\alpha x ^{2}.\) The distance at which the particle stops:

A satellite of mass \(\mathrm{m}\) is launched vertically upwards with an initial speed \(u\) from the surface of the earth. After it reaches height \(\mathrm{R}\) (\(R =\) radius of the earth), it ejects a rocket of mass \(\frac{\mathrm{m}}{10}\) so that subsequently the satellite moves in a circular orbit. The kinetic energy of the rocket is (\(G\) is the gravitational constant: \(\mathrm{M}\) is the mass of the earth)

The angle between vector \((\overrightarrow{{A}})\) and \((\overrightarrow{{A}}-\overrightarrow{{B}})\) is :

An electron of mass \(m\) is moving in an electric field \(\vec{E} = -2E_0\hat{i}\) (\(E_0 =\) constant \(> 0\)), with an initial velocity \(\vec{V} = v_0\hat{i}\) (\(v_0 =\) constant \(> 0\)). If \(\lambda_0 = \dfrac{h}{4mv_0}\), its de Broglie wavelength at time \(t\) is __________.
(\(e =\) charge of electron)

The de-Broglie wavelength associated with a particle of mass \(m\) and energy \(E\) is \(\mathrm{h} / \sqrt{2 m E}\) The dimensional formula for Planck's constant is _______.

An infinitely long current carrying wire and a small current carrying loop are in the plane of the paper as shown. the radius of the loop is \(a\) and distance of its centre from the wire is \(d (d >> a)\). If the loop applies a force \(F\) on the wire then

The electric field of a plane electromagnetic wave is given by \(\vec E = {E_0}\hat i\,\cos \,\left( {kz} \right)\,\cos \,\left( {\omega t} \right)\) The corresponding magnetic field \(\vec B\) is then given by