Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is:
Hydrogen peroxide is a chemical element including hydrogen and oxygen. Anhydrous hydrogen peroxide is a colorless, sugary liquid that breaks down quickly into oxygen and water. The chemical formula for hydrogen peroxide [H2O2] is H—O—O—H, here, each dash represents a single covalent link. Especially, each bond has a pair of mutual electrons, one from each of the atoms at the bond’s ends. The polar bonds are H—O bonds, but this is not the polar one - O—O bonds.