Given: The preparation of potassium permanganate involves two steps. In the first step, manganese dioxide (\( \text{MnO}_2 \)) reacts with potassium hydroxide (KOH) and potassium nitrate (\( \text{KNO}_3 \)) to form potassium manganate (\( \text{K}_2\text{MnO}_4 \)).
The reaction is as follows: \[ \text{MnO}_2 + 4 \text{KOH} + 2 \text{KNO}_3 \to \text{K}_2\text{MnO}_4 + 2 \text{H}_2\text{O} \] - Manganese dioxide reacts with potassium hydroxide and potassium nitrate to produce potassium manganate and water.
The product of the first step is \( \boxed{\text{K}_2\text{MnO}_4} \), which corresponds to option (4).
Let A be a 3 × 3 matrix such that \(\text{det}(A) = 5\). If \(\text{det}(3 \, \text{adj}(2A)) = 2^{\alpha \cdot 3^{\beta} \cdot 5^{\gamma}}\), then \( (\alpha + \beta + \gamma) \) is equal to: