A sample of hydrogen-like Li^2+ ions (Z=3) is excited so that every ion is in the n=3 state, and the emitted radiation is incident on a metal of work…

Each ion in $n=3$ can decay $3\to2$, $2\to1$ or $3\to1$, so three lines appear; (A) is true. The energies are $E_{31}=122.4\left(1-\tfrac19\right)=108.8\ \mathrm{eV}$, $E_{21}=122.4\times\tfrac34=91.8\ \mathrm{eV}$ and $E_{32}=122.4\left(\tfrac14-\tfrac19\right)=17.0\ \mathrm{eV}$. The most energetic photon gives $K_{max}=108.8-4.0=104.8\ \mathrm{eV}$, not $100.8\ \mathrm{eV}$, so (B) is false. For the $3\to2$ photon, $eV_s=17.0-4.0=13.0\ \mathrm{eV}$, so $V_s=13\ \mathrm{V}$ and (C) is false. The orbit radius is $r_3=0.529\times9/3=1.587\ \text{\AA}$; the quantisation condition $2\pi r_3=3\lambda$ gives $\lambda=2\pi r_3/3=3.32\ \text{\AA}$, not $9.96\ \text{\AA}$, so (D) is false. Remember that a hydrogen-like ion gives $Z^2$ times the hydrogen energies, so each level is nine times deeper than in hydrogen.