Explanation

(a) The wavelength ($\lambda)$ of lines is given by
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{\lambda}$ =$\ R\left( \frac{1}{1^{2}} - \frac{1}{n^{2}} \right)$
For Lyman series, the shortest wavelength is for $n$=$\infty$ and longest is for $n$=2.
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ \ \frac{1}{\lambda_{s}}$ =$\ R\left( \frac{1}{1^{2}} \right)\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ldots(i)$
$\frac{1}{\lambda_{L}}$ =$\ R\left( \frac{1}{1} - \frac{1}{2^{2}} \right)$= $\frac{3}{4}R$ …(ii)
Dividing Eq.(ii) by Eq. (i) , we get
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{\lambda_{L}}{\lambda_{s}} = \frac{4}{3}$
Given, $\lambda_{s}$=91.2 nm
$\Rightarrow \ \ \ \ \ \ \ \ \ \ \ \ \ \lambda_{L}$ = 91.2 $\times \frac{4}{3}$ = 121.6nm