b
Energy released for tension \(\mathrm{n}=2\) to \(\mathrm{n}=1\) of hydrogen atom \(E=13.6 Z^{2}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right)\)

\(z=1, n_{1}=1, n_{2}=2\)

\(E=13.6 \times 1 \times\left(\frac{1}{1^{2}}-\frac{1}{2^{2}}\right)\)

\(\mathrm{E}=13.6 \times \frac{3}{4} \mathrm{eV}\)

For \(\mathrm{He}^{+}\) ion \(z=2\)

\((A)\) \(n=1\) to \(n=4\)

\(E=13.6 \times 2^{2} \times\left(\frac{1}{1^{2}}-\frac{1}{4^{2}}\right)=13.6 \times \frac{15}{4} \mathrm{eV}\)

\((B)\) \(n=2\) to \(n=4\)

\(E=13.6 \times 2^{2} \times\left(\frac{1}{1^{2}}-\frac{1}{4^{2}}\right)=13.6 \times \frac{3}{4} \mathrm{eV}\)

\((C)\) \(n=2\) to \(n=5\)

\(E=13.6 \times 2^{2} \times\left(\frac{1}{2^{2}}-\frac{1}{5^{2}}\right)=13.6 \times \frac{21}{25} \mathrm{eV}\)

\((D)\) \(n=2\) to \(n=5\)

\(E=13.6 \times 2^{2} \times\left(\frac{1}{2^{2}}-\frac{1}{3^{2}}\right)=13.6 \times \frac{5}{9} \mathrm{eV}\)