Using molecular orbital theory for homonuclear and heteronuclear diatomic species (for O2 and heavier, the σ2p lies below π2p; for B2, C2, N2 and…

Fill the MOs and count. (A) $\mathrm{N_2^+}$ has 13 electrons: $\sigma_{2s}^2\sigma_{2s}^{*2}\pi_{2p}^4\sigma_{2p}^1$, giving bond order $(9-4)/2=2.5$, but the single electron in $\sigma_{2p}$ is unpaired, so it is paramagnetic: wrong. (B) Bond orders are $\mathrm{O_2^+}=2.5$, $\mathrm{O_2}=2.0$, $\mathrm{O_2^-}=1.5$ and $\mathrm{O_2^{2-}}=1.0$, and bond length varies inversely with bond order, so the stated order is correct. (C) In $\mathrm{B_2}$ the two $\pi_{2p}$ electrons occupy the two degenerate $\pi$ orbitals singly with parallel spins (Hund's rule), so $\mathrm{B_2}$ is paramagnetic with two unpaired electrons: wrong. (D) $\mathrm{NO^+}$ is isoelectronic with $\mathrm{N_2}$ (10 valence electrons, bond order 3), and has all electrons paired, so it is diamagnetic: wrong. Only (B) is correct.