Explanation:

When an electron moves from a higher energy level to a lower energy level, it emits electromagnetic radiation in the form of photons. These photons have a specific wavelength or frequency, which corresponds to a specific color or spectral line.

In a hydrogen-like sample, the energy levels are determined by the equation:

E_n = -13.6/n^2 * Z^2

where E_n is the energy of the nth level, Z is the atomic number, and n is the principal quantum number (n=1 corresponds to the ground state).

Given that an electron makes a transition from the 4th excited state to the 2nd state, let's calculate the energy difference between these two levels:

ΔE = E_4 - E_2
ΔE = (-13.6/4^2 * Z^2) - (-13.6/2^2 * Z^2)
ΔE = -0.85 * Z^2 eV

where ΔE is the energy difference in electron volts.

Now, we can use the Rydberg formula to calculate the wavelengths of the emitted photons:

1/λ = RZ^2(1/n_2^2 - 1/n_1^2)

where λ is the wavelength in meters, R is the Rydberg constant (1.097 x 10^7 m^-1), and n_1 and n_2 are the initial and final quantum numbers, respectively.

Let's plug in the values for our transition (n_1=4, n_2=2) and solve for λ:

1/λ = RZ^2(1/2^2 - 1/4^2)
λ = 6.56 x 10^-7 m

This corresponds to a spectral line in the visible region of the electromagnetic spectrum, specifically in the red part of the spectrum.

Now let's address the options:

a) 10 different spectral lines are observed
This is not true, as there is only one transition from the 4th to the 2nd level, resulting in one spectral line.

b) 6 different spectral lines are observed
This is also not true, as there is only one spectral line.

c) number of lines belonging to the Balmer series is 3
The Balmer series corresponds to transitions from higher levels to the second level (n=2). The transition we are considering (from n=4 to n=2) is indeed part of the Balmer series, so this statement is true.

d) Number of lines belonging to Paschen series is 2
The Paschen series corresponds to transitions from higher levels to the third level (n=3). The transition we are considering is not part of the Paschen series, so this statement is false.

Therefore, the correct options are b), c), and d).

4th excited state means electron are in 5th shell and 2nd state means in 2nd shell