(A) The extraction of aluminum is achieved through the Hall-Héroult process, where Al2O3 is obtained as a precipitate by bubbling CO2 through a solution of sodium aluminate, resulting in the reaction:
\(2\text{Na}[ \text{Al}(\text{OH})_4](\text{aq.}) + \text{CO}_2 \rightarrow \text{Na}_2\text{CO}_3 + \text{H}_2\text{O} + 2\text{Al}(\text{OH})_3(\downarrow) \text{ or } \text{Al}_2\text{O}_3 \cdot 2\text{H}_2\text{O} (\text{ppt})\)
(B) The electrolytic reduction of pure alumina occurs in a steel box lined with carbon (cathode), along with the presence of cryolite (Na3AlF6) and fluorspar (CaF2), which decrease the melting point and enhance the electrolyte's conductivity.
(C) In the electrolysis process of the Hall-Héroult method, graphite rods serve as anodes. At the cathode, aluminum ions are reduced:
\(\text{Al}^{3+} + 3e^- \rightarrow \text{Al}\)
At the anode, the liberated oxygen reacts with the carbon of the anode to form CO and CO2:
\(\text{C} + \text{O}_2^- \rightarrow \text{CO} + 2e^-\)
\(\text{C} + 2\text{O}_2^- \rightarrow \text{CO}_2 + 4e^-\)
(D) In this context, the cathode consists of a steel vessel with a carbon lining.
Let $ P(x_1, y_1) $ and $ Q(x_2, y_2) $ be two distinct points on the ellipse $$ \frac{x^2}{9} + \frac{y^2}{4} = 1 $$ such that $ y_1 > 0 $, and $ y_2 > 0 $. Let $ C $ denote the circle $ x^2 + y^2 = 9 $, and $ M $ be the point $ (3, 0) $. Suppose the line $ x = x_1 $ intersects $ C $ at $ R $, and the line $ x = x_2 $ intersects $ C $ at $ S $, such that the $ y $-coordinates of $ R $ and $ S $ are positive. Let $ \angle ROM = \frac{\pi}{6} $ and $ \angle SOM = \frac{\pi}{3} $, where $ O $ denotes the origin $ (0, 0) $. Let $ |XY| $ denote the length of the line segment $ XY $. Then which of the following statements is (are) TRUE?