Solution (source code)

= Solution

The C=O length is invariant under every operation, so its stretch is $A_1$. If $s_L,s_R$ are the two C–H bond-length changes, their sum and difference give
$$
Q_{\rm CO}\sim A_1,\qquad Q_{\rm CH,s}=\frac{s_L+s_R}{\sqrt2}\sim A_1,\qquad
Q_{\rm CH,a}=\frac{s_L-s_R}{\sqrt2}\sim B_1.
$$
Hence $\boxed{\Gamma_{\rm stretch}=2A_1+B_1}$. These are stretching symmetry coordinates; the two $A_1$ motions can mix in actual <normal modes>, and mass-weighted atomic motions need not have the schematic arrow lengths below.

\Image[/past-exam-of-the-chemistry-a-course-of-the-university-of-cambridge/2025/paper-1-formaldehyde-stretches.png]
{title=Original <formaldehyde> stretching symmetry sketches in the paper's $xz$ plane}