In order to shift a body of mass $m$ from a circular orbit of radius $3R$ to a higher radius $5R$ around the earth,the work done is

Match the following genes of the $Lac$ operon with their respective products:
List-$I$ (Gene) | List-$II$ (Product)
$(i)$ $i$ gene | $(1)$ $\beta$-galactosidase
$(ii)$ $z$ gene | $(2)$ Permease
$(iii)$ $y$ gene | $(3)$ Repressor
$(iv)$ $a$ gene | $(4)$ Transacetylase
Select the correct option.

After absorbing a slowly moving neutron of mass $m_N$ (momentum $\sim 0$),a nucleus of mass $M$ breaks into two nuclei of masses $m_1$ and $5m_1$ $(6m_1 = M + m_N)$. If the de-Broglie wavelength of the nucleus with mass $m_1$ is $\lambda$,then the de-Broglie wavelength of the other nucleus will be:

Three very large plates of the same area are kept parallel and close to each other. They are considered as ideal black surfaces and have very high thermal conductivity. The first and third plates are maintained at temperatures $2\,T$ and $3\,T$ respectively. The temperature of the middle (i.e.,second) plate under steady-state conditions is

The current in an $AC$ circuit is given by $i = i_1 \sin \omega t + i_2 \cos \omega t$. Its $r.m.s.$ value is:

Which of the following pairs of transition metal ions have the same calculated values of magnetic moment?