= Solution
The out-of-plane displacements in the wag transform as $y$, hence as $B_2$. A one-quantum combination of two distinct <normal modes> transforms as the product of their irreducible representations. The observed feature satisfies
$$
3950-1167=2783\ \mathrm{cm^{-1}}.
$$
A combination of the wag with an $A_1$ stretch has symmetry $B_2\otimes A_1=B_2$ and can absorb through $\mu_y$. In contrast, $B_2\otimes B_1=A_2$ has no electric-dipole component in $C_{2v}$ and is forbidden in the stated symmetry model. The missing feature near $4010=1167+2843\ \mathrm{cm^{-1}}$ therefore identifies the $2843$ stretch as $B_1$.
The completed assignments are
$$
\boxed{1746:\ \mathrm{C{=}O}\ (A_1),\quad2783:\ \mathrm{CH\ symmetric}\ (A_1),\quad2843:\ \mathrm{CH\ antisymmetric}\ (B_1),\quad3482:\ 2\nu_{\rm CO}\ (A_1).}
$$
The allowed combination obtains intensity through anharmonic/dipole-coupling terms; a purely separable harmonic model with a strictly linear dipole has no combination bands. The argument uses both the observed combination and the selection rule: absence of a feature by itself would not generally prove a forbidden symmetry.<Formaldehyde vibrational data in the NIST Chemistry WebBook>[Gas-phase fundamentals: CH2 symmetric stretch 2783, antisymmetric stretch 2843, CO stretch 1746, wag 1167 cm−1, with the same symmetry convention]{ref}
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