A compound with molecular formula C10H10O4 exhibits only

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First, notice that the molecular formula C10H10O4 indicates the hydrogen deficiency index of six. With only ten carbons in the molecule, this means we almost we almost certainly have benzene present; it would account for four elements of unsaturation (3 double bonds, 1 ring). That leaves two double bonds rings left.

The singlet at 8.1 ppm (worth four protons) is characteristic of a para disubstituted benzene with a mirror plane of symmetry. The chemical shift of the aromatic ring (slightly downfield of a typical benzene) indicates that there are electron-withdrawing groups attached to the 1,4 positions on the benzene. Carbonyl groups fit this description, and we know that we have two elements of unsaturation left. Placing a carbonyl group on the 1,4 positions of the benzene ring would leave us with two carbons, two oxygens, and six hydrogens left in the molecular formula.