Consider a hydrogen-like ionized atom with atomic number $$Z$$ with a single electron. In the emission spectrum of this atom, the photon emitted in the $$n=2$$ to $$n=1$$ transition has energy $$74.8eV$$ higher than the photon emitted in the $$n=3$$ to $$n=2$$ transition. The ionization energy of the hydrogen atom is $$13.6$$ $$eV.$$ The value of $$Z$$ is ____________.

A planet of mass $$M,$$ has two natural satellites with masses $${m_1}$$ and $${m_2}.$$ The radii of their circular orbits are $${R_1}$$ and $${R_2}$$ respectively, Ignore the gravitational force between the satellites. Define $${v_1},{L_1},{K_1}$$ and $${T_1}$$ to be , respectively, the orbital speed, angular momentum, kinetic energy and time period of revolution of satellite $$1$$; and $${v_2},{L_2},{K_2},$$ and $${T_2}$$ to be the corresponding quantities of satellite $$2.$$ Given $${m_1}/{m_2} = 2$$ and $${R_1}/{R_2} = 1/4,$$ match the ratios in List-$${\rm I}$$ to the numbers in List-$${\rm II}.$$

LIST - I		LIST - II
P.	v1/v2	1.	1/8
Q.	L1/L2	2.	1
R.	K1/K2	3.	2
S.	T1/T2	4.	8

A moving coil galvanometer has $$50$$ turns and each turn has an area $$2 \times {10^{ - 4}}\,{m^2}.$$ The magnetic field produced by the magnet inside the galvanometer is $$0.02$$ $$T.$$ The torsional constant of the suspension wire is $${10^{ - 4}}\,N\,m\,ra{d^{ - 1}}.$$ When a current flows through the galvanometer, a full scale deflection occurs if the coil rotates by $$0.2$$ $$rad$$. The resistance of the coil of the galvanometer is $$50\Omega .$$ This galvanometer is to be converted into an ammeter capable of measuring current in the range $$0-1.0$$ $$A$$. For this purpose, a shunt resistance is to be added in parallel to the galvanometer. The value of this shunt resistance, in ohms, is _____________.

The electric field $$E$$ is measured at a point $$P(0,0,d)$$ generated due to various charge distributions and the dependence of $$E$$ on $$d$$ is found to be different for different charge distributions. List-$${\rm I}$$ contains different relations between $$E$$ and $$d$$. List-$${\rm II}$$ describes different electric charge distributions, along with their locations. Match the functions in List-$${\rm I}$$ with the related charge distributions in List-$${\rm II}$$.

LIST - I		LIST - II
P.	$$E$$ is independent of $$d$$	1.	A point charge Q at the origin
Q.	$$E\, \propto \,1/d$$	2.	A small dipole with point charges $$Q$$ at $$\left( {0,0,l} \right)$$ and $$-Q$$ at $$\left( {0,0, - l} \right).$$ Take $$2l < < d$$
R.	$$E\, \propto \,1/{d^2}$$	3.	An infinite line charge coincident with the x-axis, with uniform linear charge density $$\lambda $$
S.	$$E\, \propto \,1/{d^3}$$	4.	Two infinite wires carrying uniform linear charge density parallel to the $$x$$-axis. The one along $$\left( {y = 0,z = l} \right)$$ has a charge density $$ + \lambda $$ and the one along $$\left( {y = 0,z = - l} \right)$$ has a charge density Take
		5.	Infinite plane charge coincident with the $$xy$$-plane with uniform surface charge density