Write each ion's electronic configuration beyond [Xe] and check for unpaired electrons:

La${}^{3+}$: [Xe] 4f${}^0$ → diamagnetic. Ce${}^{4+}$: [Xe] 4f${}^0$ → diamagnetic. Pair (A) is diamagnetic.

Yb${}^{2+}$: [Xe] 4f${}^{14}$ → diamagnetic. Lu${}^{3+}$: [Xe] 4f${}^{14}$ → diamagnetic. Pair (B) is diamagnetic.

La${}^{2+}$: [Xe] 5d${}^1$ → paramagnetic. Ce${}^{3+}$: [Xe] 4f${}^1$ → paramagnetic. Pair (C) is not diamagnetic.

Yb${}^{3+}$: [Xe] 4f${}^{13}$ → paramagnetic. Lu${}^{2+}$: [Xe] 4f${}^{14}$ 5d${}^1$ → paramagnetic. Pair (D) is not diamagnetic.

$\therefore$ Diamagnetic pairs are (A) and (B).