Find the radius of the first,second,and third orbits of the hydrogen atom.

For the Paschen series, the values of $n_1$ and $n_2$ in the expression $\Delta E = R h c \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right)$ are:

If the ratio of energies of an electron in the excited states of $H$ and $Li^{2+}$ is $1: 9$,the radius ratio of the electron in the same excited states of $H$ and $Li^{2+}$ is

What is the wave number of a photon emitted during the transition from orbit $n=4$ to $n=2$ in a hydrogen atom (in $cm^{-1}$)? $[R_{H}=109677 \ cm^{-1}]$

When an electron transitions from the $6^{th}$ orbit to the $2^{nd}$ orbit in a $H$ atom,what is the total number of spectral lines (excluding the Balmer series) produced?

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 $(2)$.