The emission spectrum of hydrogen is found to satisfy the expression for the energy change $\Delta E$ (in joules) such that $\Delta E = 2.18 \times 10^{-18} \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \, J$ where $n_1 = 1, 2, 3 \dots$ and $n_2 = 2, 3, 4 \dots$. The spectral lines correspond to the Paschen series when:

For a hydrogen $(H)$ atom,what will be the wavelength (in $nm$) associated with the transition of an electron from the infinite energy level to the first energy level? (Rydberg constant = $1.097 \times 10^{7} \ m^{-1}$)

Which of the following transitions emits a photon with the maximum frequency?

The energy required for the ionization of an excited hydrogen atom would be (in $eV$)-

Which hydrogen-like species will have the same radius as that of the $1^{st}$ Bohr orbit of a hydrogen atom?

Answer the following by appropriately matching the lists based on the information given in the paragraph. Consider Bohr's model of a one-electron atom where the electron moves around the nucleus. In the following,$List-I$ contains some quantities for the $n^{\text{th}}$ orbit of the atom and $List-II$ contains options showing how they depend on $n$.

$List-I$	$List-II$
$(I)$ Radius of the $n^{\text{th}}$ orbit	$(P) \propto n^{-2}$
$(II)$ Angular momentum of the electron in the $n^{\text{th}}$ orbit	$(Q) \propto n^{-1}$
$(III)$ Kinetic energy of the electron in the $n^{\text{th}}$ orbit	$(R) \propto n^0$
$(IV)$ Potential energy of the electron in the $n^{\text{th}}$ orbit	$(S) \propto n^1$
-	$(T) \propto n^2$
-	$(U) \propto n^{1/2}$

$(1)$ Which of the following options has the correct combination considering $List-I$ and $List-II$?
$(1) (II), (R)$ $(2) (I), (P)$ $(3) (I), (T)$ $(4) (II), (Q)$
$(2)$ Which of the following options has the correct combination considering $List-I$ and $List-II$?
$(1) (III), (S)$ $(2) (IV), (Q)$ $(3) (IV), (U)$ $(4) (III), (P)$
Give the answer for questions $(1)$ and $(2)$.