= Solution
First count the organic groups and then the unsatisfied <oxygen> valences. \b[A is the fully condensed <T8 silsesquioxane cage>, $\mathrm{Cy_8Si_8O_{12}}$], with <silicon> at the eight vertices of a <cube> and bridging <oxygen> on its twelve edges. Each <silicon> has one <cyclohexyl group> and three <siloxane> bridges. Its eight <cyclohexyl groups> supply exactly $\mathrm{C_{48}H_{88}}$.
\b[B is a fully condensed <T10 silsesquioxane cage>, $\mathrm{Cy_{10}Si_{10}O_{15}}$], for which a pentagonal-prismatic <silicon> skeleton with an <oxygen> on every edge is a standard compatible structure. Its ten groups give $\mathrm{C_{60}H_{110}}$. The formula establishes full condensation but does not, without more structural data, rule out every other possible ten-vertex cage connectivity.
\b[C is a corner-deficient <T7 trisilanol cage>, $\mathrm{Cy_7Si_7O_9(OH)_3}$]. Seven <cyclohexyl groups> account for $\mathrm{C_{42}H_{77}}$, leaving three hydrogen atoms as Si–OH groups. Remove one <silicon> corner of the T8 <cube> and replace the three severed connections by terminal hydroxyl groups. Nine <oxygen> atoms remain as bridges and three become terminal <silanol> <oxygen> atoms, giving precisely twelve <oxygen> atoms. There are three <silicon> sites bearing OH, three further equivalent fully condensed sites, and one opposite fully condensed site, consistent with \b[three <silicon> environments in a 3:3:1 site-count ratio]. The less negative resonance is naturally associated with the three silanol-bearing sites; the remaining two resonances belong to the two fully condensed classes. Peak heights alone are not quantitative integrations.
\Image[/past-exam-of-the-chemistry-b-course-of-the-university-of-cambridge/2021/paper-2-silsesquioxanes.png]
{title=Silicon skeletons of the T8, T10 and corner-deficient T7 cages}
The source figure appears to label the spectrum $^{27}\mathrm{Si}$. The chemically appropriate routine experiment here is \b[<silicon-29 NMR>], using the spin-one-half isotope $^{29}\mathrm{Si}$; the figure label is treated as an apparent isotope-label typographical error, not as evidence for another cage.
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